THE FOUR-COLOR THEOREMConventional proof

Franciszek Jag laNon-affiliated research

E-mail: franciszek.jagla@gmail.com

Abstract

This paper presents original concepts and methods for 4-coloring a planegraph and proving the Four-Color Theorem without the help of a computer.The graph decomposition concept is motivated by the following observation:a plane graph’s simple cycles basis can be treated as an onion by peelingcycles to form tiers. The 3-colors border concept asserts that every planegraph admits a proper 4-coloring in which the vertices of the unbounded face(border) use, at most, three colors. The concept is proven as the coloring pro-gresses. The last concept, three-color congruence, is defined for two verticesthat are assigned identical colors in every 3-coloring of a 3-colorable graph.

Interim contraction of non-significant vertices in the tiers reduces coloring topredominantly their borders coloring. One of the three border colors usedin the already colored sub-graphs is chosen as a designated color when re-cursively expanding the sub-graphs up to each tier. Vertices of the desig-nated color are temporarily removed from the sub-graphs and the enlargedgraph is expected to be 3-colorable. Color congruence methods either 3-colorthis graph or identify color conflicting vertices and resolve the conflict by re-assigning the designated color to vertices. Re-inserting the removed verticescompletes a 4-coloring of the enlarged graph. The restored non-significantvertices preserve 4-colorability in the original graph.

This direct coloring contrasts with the approach used in computer-assistedproofs, which verify that no counter-example exists to disprove graph 4-colorability.

Keywords rim, tier, non-significant vertex, pivot vertex, 3-colors border, vertexcolor congruence, 3-color conflict, refinement contour.

2010 Mathematics Subject Classification: 05C10, 05C15.

1 Introduction

The problem of 4-coloring a graph was introduced in 1852 by Francis Guthrie [8] formap coloring. The original topological problem can be converted into a combinato-rial one [6], and a vertex four-color assignment is a preferred problem formulation.

This paper focuses on assigning colors to the vertices of a plane graph with thegoal of proving the Four Color Theorem without the help of a computer.

We begin with the proof overview followed by a brief history of the Four ColorTheorem computer-assisted proofs. The last sub-section provides assumptions forthis work and expands on standard conventions.

1.1 Four Color Theorem proof overviewThe “Graph decomposition” concept prepares fundamental “Rims and Tiers” datastructures, the starting point to all other graph structural concepts and additional

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data structures. The “3-colors border” and “Vertex 3-colors congruence” conceptsprovide a 3-coloring foundation. These three concepts are prerequisites for coloringalgorithms and they are discussed in Sections two through four.

ConceptsA planar graph can be embedded in the plane and its cycle space has a simple basis[6]. “Graph decomposition” (2.1) partitions a simple cycle basis for the graph andprepares data structures: rims (2.1.1) or subsets of edges, and tiers (2.1.2) or subsetsof partitioned basis cycles. A rim is the walk of an unbounded face of the currentlyanalyzed graph or current set of basis cycles and a tier is a subset of current setbasis cycles with at least one vertex in its corresponding rim. Specifying a tiernaturally defines its descendant tiers and its parent tier when it is distinct fromthe first derived tier (root tier). Tiers derivation dependencies are best expressedby the decomposition tree (2.2). Three important ideas related to decompositionare: “non-significant vertices” (2.3), “parts of a tier” (2.4), and “pivot verticesof a path between rim and its sub-rim vertices” (2.6). The contraction of non-significant vertices reduces coloring to predominantly rim coloring. The breakdownof a tier into intuitively defined parts or subsets of cycles (Polygona, Tweel, andShell) further divides and reduces coloring complexity. A pivot vertex plays essentialrole in color-conflict avoidance analysis.

Our graph decomposition is elaborate but tiers are indispensable for coloringin comparison to the layers received by partitioning the graph’s vertices (k-outer-planar graphs) as defined by Baker [3]. Layers other than the first one are not simplecycles, with clean separation of a layer from other layers. Layers do not allow foreasy breakdown into parts, but the real drawback is impossibility in using 3-coloringalgorithms. Other types of decomposition are not suitable for graph coloring exceptin special cases, such as the decomposition of planar perfect graphs[12].

The “3 -colors border” concept asserts that every plane graph admits a proper 4-coloring in which the vertices of the unbounded face use, at most, three colors.The fixed color (3.1.1) is the fourth color not to be used by the vertices of theunbounded face. The concept has been added to the definition of the four-colorproblem as requirement 3.1.D2 for a graph to be 4-colorable, and is verified as thecoloring process progresses. This concept, when combined with contraction of non-significant vertices, “allows for reducing the coloring scope from four to three colorsand allows for applying 3-coloring algorithms”.

The concept “Vertex three-color congruence” (4.1.1) is of primary importance in a3-colorable graph. Two vertices in a graph are color-congruent if the colors assignedto them are identical in every 3-coloring of the graph. The color congruence meth-ods are inspired by Fisk’s work [7]. Using the 3-coloring rules 4.2.E, the congruencecoloring algorithm (Lemma 4.3.1) 3-colors a 3-colorable graph. The coloring algo-rithm can be used to identify a color collision of two vertices in a sub-graph (4.5.3),which is 3-colorable when the edge of the color colliding vertices is removed fromthe sub-graph. The “3-coloring algorithm” and the definition of “color conflict” arethe main results of this section.

A concept comparable to the “vertex three-color congruence” is the “four-colorvertex fixation” specified in a draft paper by Brændeland [5].

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Algorithmic proofThe non-significant vertices are contracted and coloring begins with the innermostleaf tiers. The already 4-colored sub-graphs are recursively expanded up to thenext tier and the enlarged graph is proven to be 4-colorable. Usually, “Fixed colorselection” mechanism suffices to color the enlarged graph. In the case of a collision,“Fixed color refinement” resolves the conflict. Once the sub-graphs are expandedup to the root tier, the entire graph becomes 4-colored. The non-significant verticesare re-inserted and assigned colors, resulting in a 4-coloring the original graph. Theproof spans Sections three and five through eight, and the Sections are organizedby the depth level of the graph’s decomposition tree.

Graph is one level deep. The only leaf tier in a graph contains a mix of Polygona(2.4.1) and Tweel (2.4.3) parts. The parts are colored and their colors are synchro-nized by observing 2.5.C2 Property. Theorem 3.2.1 summarizes the result: “Onelevel deep graphs are 4-colorable”.

Graph is two or more levels deep - fixed color selection. When expanding 4-colored sub-graphs to the next tier, we designate the fixed color to be the colorof the selected subset of the sub-graphs’ border vertices. This color is not to beused by the next tier’s border. “A sub-graph induced by the next tier with omittedvertices of the fixed color is 3-colorable” is the main result of Lemma 5.2.1. Weexpect the enlarged graph with absent fixed color vertices to be 3-colorable.

Graph is two levels deep. Fixed color selection does not guarantee 4-colorabilityof a two or more levels deep graph. The Fritsch [8] graph breaks this expectation.Its edges between the inner rim vertices form a “triangle-cycle” (6.1 Tarc) and twoof the triangle vertices are congruent regardless which color has been selected asthe fixed one. Avoiding a complex analysis to determine which graphs are or arenot 3-colorable, we choose a pragmatic approach: graphs that contain a “triangle-cycle” are easily recognized and can be efficiently 3-colored as shown in Lemma 6.1.2.Theorem 6.3 summarizes the main result: “Two levels deep graphs are 4-colorable”.

Graph is two or more levels deep - fixed color refinement . Assume the fixed colorhas been selected. The 3-coloring algorithm (4.3.1) may fail to 3-color an enlargedgraph with omitted fixed color vertices. A conflict may also occur when re-insertingnon-significant vertices. Let identify a conflict sub-graph (4.5) and an embeddedwithin it a closed path (“refinement contour” 7.1). We propose a solution “Fixedcolor refinement” which re-positions the fixed color vertices inside the refinementcontour to make the conflict sub-graph 3-colorable. The main result “Fixed colorrefinement resolves conflicts” is declared in Lemma 7.4, and Proposition 7.5 statesthat “The original graph with omitted non-significant vertices is 4-colorable”.

Graph is of any level deep - restoring non-significant vertices . Beginning with theinnermost tiers, the non-significant vertices suppressed by a contraction rule are re-inserted and colored. These vertices are verified whether they can be colored withoutmodifying colors of neighboring vertices (Algorithm 8.1.1). When the Algorithmfails, we try first 2-color exchanges. In the worst case, the graph with omitted fixedcolor vertices is re-3-colored, and fixed color refinement is applied to resolve theconflict. The important result “Restored non-significant vertices preserve graph’s4-colorability” is stated in Lemma 8.1.2.The above algorithms 4-color directly a planar graph, proving the Four-Color Theo-rem. Proposition 8.2 summarizes the final result: “A planar graph is 4-colorable”.

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1.2 Formal computer methods as a principal proof techniqueUntil the second half of the twentieth century, it was assumed that the validity ofany proof could, in principle, be checked by a competent mathematician. A proofgiven by Kempe [14] in 1879 turned out to be flawed. The seeming impossibilityof proving the Four-Color Theorem and the increasing power of computers in theearly 1960’s led mathematicians to use computers to show that there are no counter-examples to 4-colorability. Their approach [8] was to search for “an unavoidable setof reducible configurations” where an unavoidable configuration is reducible for a4-colorable graph if it cannot occur in a minimal graph failing 4-colorability. Bothunavoidability and reducibility concepts can be traced back to Kempe [14].

In 1904, Wernicke [8] introduced the discharging method to prove a theorem,which was part of an attempt to prove the Four-Color Theorem. Heesch [8] useddischarging ([8] 7.2) for the four-color problem, which turned out to be importantin the unavoidability portion of the subsequent Appel-Haken proof. Heesch alsoexpanded on reducibility, the concept coined by Birkhoff [8] in 1913.

As a result of massive manual and computerized work, the Four-Color Theoremwas proven in 1977 by Appel and Haken [2] (A-H), assisted in algorithmic work byKoch ([2]). It was the first major theorem proven with the help of a computer. Ini-tially, the proof did not gain general approval among mathematicians, but did fuela debate[16] on the meaning of a mathematical proof. Because of the size and com-plexity of the human-verifiable portion and considerable uncertainty about the A-Hproof, a proof that is simpler in several aspects, using the same ideas and still relyingon computers, was published in 1997 by Robertson, Sanders, Seymour, and Thomas[15]. The more recent proof uses 633 (A-H: 1476) unavoidable configurations and 32(A-H: 487) discharging rules. The non-computational parts are verifiable manually.Reducibility was tested by independent programs and unavoidability confirmed byhand-checkable and formally written proofs validated by a computer. This proof ispredominantly accepted. In 2005, the theorem was proven by Gonthier [10] witha general-purpose theorem-proving software. The formal proof specification followsthe same main ideas as Robertson, Sanders, Seymour, and Thomas.

These proofs helped pioneer the development of formal methods as a principal prooftechnique and increased the acceptance and importance of formal mathematics.

The history of the four-color problem, including the contributions of mathemati-cians, the topological background and the computer-assisted proof, can be found inFritsch & Fritsch [8]. An in-depth review of plane graph colorings can be found inBorodin’s work [4].

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1.3 PreliminariesFor standard graph terminology, notation, and conventions, we refer to Diestel [6].

1.3.1 AssumptionsA plane graph G ([6] pp. 70-74) comprises a set of vertices V and a set of edges E ;G = (V , E ). We assume the graph G is at least two-connected and is undirected;it has no loops, bridges, cut vertices, and parallel edges, and each vertex has degreegreater than two. A simple cycles basis of the graph G is available ([13]).

1.3.2 Additional Conventions and Notation

We introduce in the subsequent sections several novel concepts, non-standard def-initions, functions, and operations, along with some associated notation. A non-standard definition is distinguished with the use of boldface term and by a sym- term

bol/term as shown in the margin here. A sub-section number in the margin locatesthe definition of a referenced term/symbol. Beginnings and ends of algorithms and 1.3.2

ends of proofs are marked respectively by the symbols (3, �) and �.

Bold lines in diagrams of figures represent rim’s edges. A circle stands for avertex, and an enclosed digit is the color assigned to it; a square box keeps thefixed color. Usually vertex identifier is placed next to its drawing, occasionally it isplaced inside its drawing. Deviations from these conventions are always explained.

Because the terms “cycle” and “circuit” are used repeatedly, we briefly definethem. A cycle is a closed path of distinct edges. A simple cycle contains verticesthat belong to exactly two edges.

A cycle basis is a maximal set B of independent simple cycles, that is, a cycleC∈B cannot be a linear combination of cycles in B\C . A simple cycle basis consistsof simple cycles where each simple cycle is a walk around some inner face of thegraph G and each edge belongs to one or two basis cycles (MacLane’s planaritycriterion). Please refer to [6] (pp. 20-24, 68-72, 85) for details.

In this paper, a circuit is a simple cycle and a cycle is a member of B. circuit

cycleBelow are some frequently used standard and non-standard notation for the

elements of a graph G .N (v) the set of vertices adjacent to vertex v∈V

d(v) the degree of vertex v

B() derives a simple cycle basis B for a graph G ; for example B = B(G)

V() the set of vertices contained in a set of edges, cycles, rims, tiers, or graphs;for example V = V(G)

E() the set of edges contained in a set of cycles, rims, tiers or graphs;for example E = E(G)

G() the graph induced by a set of edges, cycles, or tiers; G = G({e}∪{C}∪{T })ξe, µe the end vertices of an edge e∈E

NC(v) the cycle neighborhood of vertex v contains all basis cycles that include v

NC(v) = {C : C∈B∧v∈V(C )}

X (A) the edge border of base cycles A⊆ B

X (A) ={

e: e∈ E(A) ∧ ( | {C : C∈ A ∧ e∈ E(C )} | = 1)}

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2 Plane Graph Decomposition

This section presents an algorithm for decomposing a single component of a planegraph into rims and tiers. Furthermore, three concepts relevant to coloring aredefined: non-significant vertices, tier parts, and between-rim paths.

2.1 Plane graph decomposition into rims and tiers

Given a cycle basis B = {C j} (j = 1, . . . , | E | - | V | + 1) for G , we specifya decomposition of B into rims {Rx} (2.1.1) or subsets of edges and partition Binto tiers {T x} (2.1.2) or subsets of cycles using decomposition indices {x}. Theindex x is initially set to 1, and increases its value (x .i , i = 1, . . . ) when derivingconsecutive rims and tiers from the current set of cycles Bx (Bx ⊆ B).

Decomposition algorithm3 Consider a stack S = {(x j, Bxj

)}(for j =1,. . . ) of entries where each entrycontains a decomposition index and a set of cycles. The stack S is initialized to{1,B}. In each iteration of the decomposition, the top entry in the stack, (x , Bx)is popped off and the corresponding rim and tier are determined.

2.1.1 RimA rim Rx is the set of edges of an unbounded face of a plane graph represented byrim Rx

the set of cycles Bx:

Rx = X (Bx)2.1.2 TierA tier T x is a set of these Bx cycles, in which at least one vertex belongs to thetier T x

rim Rx:

T x = {C : C∈Bx∧ V(C )∩V(Rx) 6= ∅}Partition reduced set Bx

For the reduced set B∗x = Bx\T x, one of two cases can occur:

• B∗x is empty. If the stack S is empty, then decomposition is completed.

Otherwise, processing resumes with the top entry of the stack.

• B∗x is partitioned into k (k ≥ 1) edge-disjoint subsets (B∗x.1, . . . , B∗x.k):if k ≥ 2 then, ∀i , j (∀C 1∈B∗x.i (∀C 2∈B∗x.j (E(C 1)∩E(C 2) = ∅) ) ) 1≤i 6= j≤k ,and if |B∗x.i| ≥ 2 then, the cycles of a subset share edges:∀C 1∈B∗x.i (∃C 2∈B∗x.i (E(C 1)∩E(C 2) 6= ∅) ) where C 1 6=C 2, 1≤i ≤k .For each i (i = 1, . . . , k) a pair (x .i , B∗x.i) is pushed to the stack S .

Processing resumes with the top stack entry. �

Figure 1: Graph decomposition into rims and tiers

Proposition 2.1.3 Partitioning a plane graph’s cycle into tiers is well- defined.Proof . At each step of the decomposition, the set Bx produces a new tier T x andthe reduced non-empty set B∗x is partitioned into subsets {B∗x.i}. In the recursiveprocess each subset B∗x.i produces a tier and eventually is reduced to an empty set.�

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2.2 Associated elements of the decompositionThe relation y≺x means that the tier T y is directly preceded by the tier T x in ≺the decomposition process. This relation is asymmetric and intransitive. The setof decomposition indices {x , . . . } with the “≺” relation induces a tree TG with TG

its root equal to 1. This tree is termed a decomposition tree. Within TG werefer to the existing preceding rim(tier) as the sup-rim(sup-tier), and to an existingdescending rim(tier) as a sub-rim(sub-tier).

The rims and tiers have the following noticeable properties:2.2.A1 A rim contains at least three vertices.

Two vertices in a rim would form parallel edges.2.2.A2 No two rims Rx and Ry share a vertex or an edge, where y≺x .2.2.A3 Two or more sub-rims Ry,. . . ,Rz may share a vertex, where y≺x ,. . . ,z≺x .2.2.A4 The internal vertices I x do not form a circuit.

This circuit would either generate a new tier or modify some of the tiers.

We give some common and non-standard definitions related to the decomposi-tion. These definitions are necessary for other concepts and algorithms.

2.2.1 Root and leaf rims/tiersThe rim/tier derived as the first one is termed the root rim/tier. A leaf rim/tier root

is a rim/tier (Rx/T x) for which the subset Bx reduced by the tier T x is empty. leaf

2.2.2 Gx and GkD graphs

For the subset Bx (x∈TG), let the graph Gx be defined by Gx = G(Bx). The graphGx includes the tier T x and all recursively descendant sub-tiers of T x. Gx

A graph G (G∈GkD) has the depth of its decomposition tree TG equal k . Gk

D2.2.3 Internal I x and E x setsThe sets I x and E x denote the internal vertices and the internal edges of thetier T x, defined as follows:

I x = V(T x) \ (V(Rx) ∪ V(⋃

y≺xRy) ) I x

E x = E(T x) \ (Rx ∪⋃

y≺xRy) Ex

2.2.4 Arcs and polygonsAn arc is an internal edge u (u∈E x) incident to rim vertices: u = st ; s , t ∈V(Rx). arc

A polygon is a cycle C∈T x satisfying the following two conditions: polygon

1 C includes at least one arc2 C does not include an internal vertex in I x

A polygon usually includes at least one rim edge, although it may contain only arcs.

2.2.5 Triangles and triangulaA triangle is a cycle or a circuit with three vertices. triangle

A triangula T is a set of one or more triangle cycles sharing edges and the union triangulaof its cycles induces a 2-connected subgraph.

T = {Ti}, Ti∈B for 1≤i≤k , k ≥ 1 T

Triangles and triangula are independent of the decomposition. The terms Triangulaand Polygona mean “triangles” and “polygons” in Latin.

2.3 Non-significant verticesThe purpose of contracting non-significant vertices is to simplify the graph andreduce the coloring problem to mostly rim vertices. Non-significant vertices areexclusively internal vertices I x of the tier T x. Let a set I v (I v⊆I x) consist of

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adjacent internal vertices where a graph GIv induced by I v vertices is 1-connected.Let a set V v be the rim vertices (V v ⊆ V(Rx)) reachable from I v vertices. Theset V v is not restricted to a contiguous subset from the rim Rx; between two rimvertices in V v, there may exist an Rx vertex not adjacent to an I v vertex.

Non-significant vertices are contracted or removed by the following rules.non-significant Initial The graph GIv is contracted to a single vertex iv (iv∈I v) while preserving

its adjacency to V v and sub-rims vertices.2.3.B1 Vertex iv adjacent to two V v vertices is reduced to an edge of V v vertices.2.3.B2 Vertex iv adjacent to a sub-rim vertex v (v∈V(Ry)) is contracted to v and

v becomes adjacent to V v vertices.2.3.B3 Vertex iv is adjacent to vertices of two or more sub-rims and optionally to

V v vertices. The Initial contraction suffices.2.3.B4 At least three V v vertices are adjacent to iv. The Initial contraction suffices.

Figure 2: Contraction of non-significant verticesThe non-significant vertices use black circles, the contracted use darker ones.By the rule 2.3.B1, two adjacent internal vertices which are adjacent to two rim’svertices in Fig. 2.A, are reduced to an edge in 2.B. Figs. 2.A and 2.B also illustratethe rule 2.3.B4. The rules 2.3.B2 and 2.3.B3 are shown in Figs. 2.C and 2.D.

Non-significant Vertices AssumptionFrom now on, there are no non-significant vertices present in the modified graph G .The graph G still satisfies the assumptions 1.3.1 and its cycles, tiers, as well as theother elements associated with the decomposition are modified respectively.

2.4 Tier partsThe cycles in a tier can be divided into not necessarily disjoint subsets of cyclescalled parts. The union of a part’s cycles induces a 2-connected graph. We definethe following types of parts.

2.4.1 PolygonaA Polygona P is a maximal subset of T x polygons sharing arcs.Polygona

P = {pi}; pi∈T x for 1≤i≤k , k ≥ 1P

Polygona does not contain internal vertices; its vertices belong to the border X (P).

2.4.2 ShellA Shell Sy is a subset of T x cycles encircling sub-rim Ry (y ≺ x ) where each Shell’sShell

cycle shares a vertex with the sub-rim Ry.Sy = {C : C∈T x ∧ V(C )∩V(Ry) 6= ∅}Sy

2.4.3 TweelConsider a maximal subset T of internal vertices (T = {t1, . . . , tk}, T⊆I x) whoseTweel

neighborhood cycles {NC(t i)} share a cycle. The neighborhoods’ cycles are referredto as a Tweel.

T = {NC(t1 ), . . . , NC(tk)}Vertex ti (ti∈T ) cannot be adjacent to a vertex of one Shell because of the rule

2.3.B2, but it can be adjacent to vertices of two or more Shells and to rim’s vertices

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(2.3.B3). Finally, vertex ti can only be adjacent to rim’s vertices (2.3.B4). Becauseof these differences we will distinguish two Tweel instances.

TRim - none of the T vertices is adjacent to a vertex of a sub-rim in T x. TRim

TShl - exists at least one vertex ti (ti∈T ) that is adjacent to vertices of at least TShl

two distinct sub-rims in T x.A leaf tier T x with no internal vertex and no arc is classified as a Tweel TRim.

Note that TRim or Polygona part may occupy the entire leaf tier.

2.5 Analysis of tier’s partsTwo parts are linked if exist two adjacent internal vertices of both parts. link

A maximal group of linked Shells or Shells sharing an internal vertex is denotedby MY . The union of MY Shells induces a 2-connected sub-graph.MY = {Sy: y∈Y } where Y = {y : y≺x and ∀y1∈Y ∃y2∈Y (y2 6=y1) that MY

|V(Ry1)∩V(Ry2)| = 1 ∨ (∃e=st (s∈V(Ry1), t∈V(Ry2))) }

Figure 3: Tier partsFigure presents all types of parts. Shell symbols are placed within encircled sub-rims. MY contains S1 and S2 Shells. Vertices of S1 and S2 sub-rims, vertex p ofTShl, S3 and S4 sub-rims, vertex q of TShl, and S5 sub-rim satisfy Property 2.5.C3.

Two parts are neighbors if exists one or twoRx’s vertices adjacent to V(T x)\V(Rx)vertices of both parts. The following properties are important for coloring.2.5.C1 Leaf tier consists of TRim and Polygona parts.By its definition, a leaf tier does not contain a Shell part.

2.5.C2 Two rim vertices are adjacent to vertices of two neighboring parts.Proof . Two parts separated by an arc share two arc’s vertices. A Polygona neigh-boring with another part also results in two shared rim’s vertices. A Polygona mayshare an Rx vertex with another Polygona, but they are not neighbors.

A Shell and a neighboring another Shell or Tweel part, and two rim verticesadjacent to vertices of both parts already form a circuit. A third Rx vertex adjacentto vertices of both parts would cross this circuit. �2.5.C3 Linked TShl with MY or linked with Shells Sy parts form a tree.Proof . The parts via its adjacent vertices do not form a circuit because they wouldinduce a non-existing rim. �2.6 Path Px,Y and pivot vertices Pivx,Y

We first define two useful functions. Let up(s) be a subset of the end vertices ofthese edges E x whose other end vertices are equal to a sub-rim Ry vertex s (y≺x ),and let down(t) be a subset of the end vertices of these edges E x that are not arcsand whose other end vertices are equal to a rim Rx vertex t .

up(s) = N (s) ∩ {p: ps∈E x} where s∈V(Ry) up()

down(t) = N (t) ∩ {p: pt∈E x ∧ p /∈V(Rx)} where t∈V(Rx) down()

Let Y = {y1 , . . . , yn} be a sub-set of shells, for i = 1, . . . , n: yi≺x where n ≥ 1.

A path Px,Y consists of tier T x vertices: Px,Y = (v 1, . . . , vk) where k ≥ 3, v i−1v i∈E x Px,Y

for 1<i≤k , and the consecutive path vertices alternate between the rim Rx and the

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sub-rim Ry vertices (y∈Y ). For each y∈Y at least one Ry vertex must be presentin Px,Y . Assume v i∈V(Rx), v i−1∈V(Ry) and let derive:

V vi = down(v i) ∩ V(Ry)V vi−1

= up(v i−1) ∩ V(Rx)

Let both subsets have their vertices in order along the rim they belong to.Obviously, v i∈V vi−1

and v i−1∈V vi . Vertex v i or v i−1 must be either the first or thelast one in the corresponding subset (Fig. 4.A).

Besides, we require the path Px,Y be maximal, that is, if a vertex would extenda path then the vertex is included into the path. Finally, a vertex belongs to onepath Px,Y , but it can belong to another path Px,Z between rims Rx and Rz (z ∈Z , Y 6= Z ). On the other hand, two neighboring shells Sy1 and Sy2 may have twodifferent closed paths (Fig. 4.B)

2.6.1 Closed pathA path Px,Y is closed if v 1 = vk.

A path Px,Y is semi-closed if v 1 6= vk, its end vertices (v 1, vk) belong to the samecycle of four or more vertices, and within this cycle two path’s vertices belong tothe same rim and are adjacent.A path that is neither closed nor semi-open is termed open.

Only one closed or one semi-closed path exists between rims Rx and sub-rims{Ry} for y∈Y , while open paths may occur several times. The closed path is ofprimary importance.

2.6.2 Pivot verticesWe define pivot vertices Pivx,Y for a path Px,Ypivot

Pivx,Y Pivx,Y = V(Px,Y ) ∩ (⋃

y∈Y V(Ry))

We refer to pivot vertices briefly as pivots. A pivot must not be the first or thelast vertex in a semi-closed or open path Px,Y .

Figure 4: Path Px,y and pivots Pivx,y

Fig. 4.A illustrates a requirement for adjacent vertices vi−1 and vi (darker circles) inthe path Px,y. In Fig. 4.B, dashed lines show paths: closed Px,y1 , semi-closed Px,y2

and Px,y3 , and two Px,y2 open paths. Note there exists a closed path Px,y2y3 whilethere is no individual closed paths for the Shells Sy2 and Sy3 . There are two differentpaths Px,y1y2 for the Shells Sy1 and Sy2 . Pivots are marked as darker circles.

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3 Coloring Basics and G1D Graphs

The section introduces color and coloring-related concepts and algorithms applicableto G1

D graphs. 2.2.2

3.1 ColorsThe four colors of the palette Π are denoted by the numbers {1, 2, 3, 4} or by the h

symbols {c1, c2, c3, c4}. The set h consists of two colors: h = {c1, c2}. The h+ set h+

contains three colors: h+ = {c1, c2, c3}. Two color operations are defined as follows.

c(v) The color assigned to vertex v , c(v)∈Π c()

K(P) The set of colors assigned to the vertices of a set P : P⊆V , K(P)⊆Π K()

3.1.1 3-colors borderThe 3-colors border concept asserts that every plane graph admits a proper 4- 3-colors

bordercoloring in which the vertices of the unbounded face use, at most, three colors.The fourth color, which is not used in the graph’s border, is called the fixed fixed cF

color and is denoted by cF .

3.1.2 Four-coloringA four-coloring of a planar graph is a mapping c: V→ Π such that:3.1.D1 ∀e∈E (c(ξe) 6=c(µe)); the end vertices of each edge e have different colors.3.1.D2 |K(V(Rroot))| ≤ 3; the vertices of the root rim require at most three colors.

The criterion 3.1.D2 means that only the root rim vertices do not use the cFcolor, while all other vertices may use all Π colors. The criterion 3.1.D2 is provenas the coloring progresses. The terms “proper 4-coloring” or “4-coloring” referto graph coloring satisfying both criteria.

3.2 Leaf tier coloring 2.2.1

A leaf tier is a G1D graph and consists of Polygonas and TRim Tweels (Property 2.4.1

2.4.32.5.C1). Let us recall Fisk’s 3-coloring algorithm [7] applicable to Polygonas.

Fisk’s algorithm3 Select a cycle C in the Polygona and select two adjacent vertices p, q∈V(C ),pq∈E(C ). Assign the first two colors in h+ to p and q . The next two steps arerepeated until the entire Polygona is colored.1 Color the remaining vertices in the cycle C with the colors in h+. A vertex in atriangle is assigned the available third color. A cycle with more than three verticescan be triangulated, and the imaginary triangles are successively colored.2 Choose a new cycle C to be colored, where C shares an edge with an alreadycolored cycle. �

Theorem 3.2.1 Graphs G1D are 4-colorable.

Proof . Suppose that the root tier of a graph G (G∈G1D) contains a single part,

either a Polygona or a TRim. Fisk’s algorithm 3-colors the vertices of a Polygona.The border of a Tweel TRim coloring requires two colors and the third color isneeded when the number of the border vertices is odd. TRim’s internal vertices, oneor more non-adjacent to each other, are assigned the Π \h+ color.

Let the leaf tier contain a mix of Polygonas and TRims (Property 2.5.C1). Tocolor the tier, select any part type (either a Polygona or a TRim) and color thispart. The next selected part for coloring shares two rim’s vertices (2.5.C2) with thecurrent part. The colors of shared vertices initiate the coloring in the next part.

The leaf rim requires at most three colors and the graph G is 4-colorable. �

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4 Vertex Color Congruence

Vertex three-color congruence is defined and the 3-colorability of a graph is analyzedwith subsequent important remarks on color conflicts in a graph that requires morethan three colors. Graph H satisfies the assumptions specified in Section 1.3.1,although vertices of degree two are allowed. Let the three colors be {c1, c2, c3}.3.1

4.1 Vertex color relationsThe colors of two vertices s and t (s 6=t) in a 3-colored plane graph H can be inone of the three relations: congruence, anti-congruence, and independence. Colorcongruence is the primary relation of interest.

4.1.1 Vertex color congruenceTwo vertices s and t are color congruent if they are assigned the same color incongruence

every 3-coloring of H .

The other two relations are defined as follows.Vertices s and t are anti-congruent, if c(s) 6= c(t) in every 3-coloring of H .Vertices s and t are independent if t can be assigned any color given c(s); thatmeans in one 3-coloring vertex t is assigned a color different from c(s) while inanother 3-coloring vertex t is assigned the same color.

An edge is a trivial example of anti-congruence. Non-trivial examples of con-gruent and anti-congruent vertices were first presented in Havel’s work [11].

Figure 5: Vertex color relationsIn Fig. 5.A a congruence occurs for two triangle cycles sharing an edge (known asdiamond graph) whose two colors 2 and 3 determine the same color 1 for verticess and t . Example of anti-congruence is shown in Fig. 5.B, a subgraph of graph inFig.3 [1]. The anti-congruence embraces the quasi -edge concept defined in [1].

4.2 Congruence 3-coloringAssume the vertices of a 3-colorable graph H have not been colored yet. The goal ofthe congruence 3-coloring is to assign colors to the graph’s vertices using one or moresteps of congruence-coerced coloring. In each coloring step, colors are assigned tonon-colored yet vertices by repetitively executing one of the following rules: 4.2.E1,4.2.E2, and 4.2.E3.

4.2.1 Congruence coloring rulescoloring

rules Let us denote the already colored graph vertices by V C and the non-colored verticesby V N (V N = V(H ) \V C). One or more vertices in VN are assigned colors.4.2.E1 A vertex v (v∈V N) is adjacent to two vertices {s , t} of distinct colors;

c(v) ::= h+\K(s , t). This rule is a generalization of Fisk’s coloring rule.4.2.E2 A triangula vertex in V(T ) is the only colored vertex. Potential coloring

is the OneSlot case in Lemma 4.2.2.4.2.E3 A triangula T of vertices in V N has at least two vertices adjacent to one

or more V C vertices. Potential colorings are discussed in Lemma 4.2.2.

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Lemma 4.2.2 Three-coloring analysis of non-colored triangula vertices adjacentto at least two colored vertices results in either an option of two colors, a colorcollision, or coloring.Proof . Assume a triangula T contains non-colored vertices V(T )⊂V N , adjacentto a subset of already colored vertices V c(V c⊂V C).

Partition the triangula T vertices into three subsets {S 1, S 2, S 3} of congruentvertices, or color-slots, and determine the available colors H+\{K(V j

c)} for theV j

c vertices adjacent to S j vertices. Let a(i , j ) be an element of a matrix repre-senting color availability (1-0 value), where the i th row represents the color and thej th column indicates a color-slot (1 ≤ i , j ≤ 3). A permutation π is over (1 , 2 , 3 )indexes. Four cases are possible.Option There exist two permutations π1, π2 (π1 6= π2) such that the values

a(i , πi1), a(i , πi

2) are all 1. No vertex is assigned a color.Collision There does not exist a permutation π such that the values a(i , πi)

are all 1.ThreeSlots There exists only one permutation π such that the values a(i , πi)coloring are all 1. In this case, all vertices in T are assigned colors.OneSlot There exist two permutations π1, π2 (π1 6=π2) such that the valuescoloring a(i , πi

1), a(i , πi2) are all 1, and exists j (1≤ j ≤ 3) such that πj

1=πj2.

In this case, all vertices in T of the j th slot are assigned the πj1 color.�

Figure 6: Color slots (a, b, c) in triangulaThe diagrams relate to Lemma 4.2.2 cases: A - Option, B - OneSlot , C - Collision.

Color congruence is limited to and exclusively used for vertices colored in onecoloring step by executing the rules 4.2.E. Some of the congruent vertices thatrequire more than one coloring step can be determined (for example Fig. 1 in [11]),but this knowledge is not essential to 3-color a 3-colorable graph or to determinewhether a graph expected to be 3-colorable is not 3-colorable.4.2.3 Unary and binary congruences, and enabling verticesLet s and t vertices belong to two triangles sharing an edge pq . Assume s has been unary

congru-ence

assigned a color. Then vertex t is assigned the s ’s color in the same coloring step.This frequently appearing congruence (Fig. 5.A) is termed a unary congruence.The {p, q} vertices that allow a congruent vertex be derived are termed enabling enabling

verticesvertices. A congruence in which a vertex is derived by OneSlot mechanism or fromtwo congruent vertices is termed binary (example: vertices ’3 ’ and ’a’ in Fig. 6.B). binary

4.3 Congruence coloring and 3-colorabilityWe present a 3-coloring algorithm. The standard backtracking mechanism and thecoloring rules 4.2.E are at the heart of the algorithm. The backtracking term wascoined by Derrick Henry Lehmer in the 1950s.

Lemma 4.3.1 The congruent coloring algorithm colors a 3-colorable graph.Proof . Assume a 3-colorable graph H contains at least one triangle cycle C . By atheorem of Grotzsch [6] (p. 97), a planar graph is 3-colorable if there is no trianglein the graph. This result has been extended in many ways (see [4] for examples).

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3 We begin the coloring with a triangle cycle C that possibly one of its vertices pbelongs to the border. We assign color to p (c(p) = c1; V C = {p}; V N = V(H )\{p})and set its two additional attributes: “choice number” r = 1, and “alternate color”to “not available”. The choice numbers simplify backtracking.

Coloring step. The rules 4.2.E are executed in any order and vertices assigned colorsreceive a reference to the choice number r . The only exception is rule 4.2.E2 takingprecedence over 4.2.E3. The step ends when one of the following cases occurs.

1 All vertices have been assigned colors. The coloring terminates successfully.

2 Coloring step results in Collision.All vertices with the last choice vertex number r are reset to their initial values.If the alternate color for the choice vertex r has not been used yet, then this coloris assigned to r with no alternate color available, and processing resumes with thecoloring step. If both colors for the choice vertex have been tried, then the algorithmbacktracks to the previous choice r - 1 vertex and repeats the analysis.

3 Coloring step results in Option or no rule can be executed.Select a new choice vertex adjacent to a colored one that belongs to a cycle withthe fewest non-colored vertices. Increment the choice number r by one. Assign thecolor to this vertex with the lowest available index ci∈h+ and assign the alternatecolor to the highest available index ci∈h+. Continue with the coloring step. �

When back-tracking reaches the initial vertex p with choice number r = 1, thatmeans all the possible 3-colorings have been tried and failed. This contradicts theassumption that H is 3-colorable and exists a 3-coloringH 3Clrng.

Now suppose the graph H 3Clrng is the only 3-colored instance of the graph H .We show that the algorithm 3-colors the graph H and the colors assigned are apermutation of the colors used for H 3Clrng. The algorithm begins with a triangle inH and the assigned colors to two adjacent vertices correspond to a permutation oftwo colors in H 3Clrng. We apply the coloring steps. Suppose the coloring fails fora vertex and its selected color. The algorithm backtracks to this vertex and color-ing resumes with the alternative color which corresponds to the permuted color inH 3Clrng, whereas the colors of the previous choices equal to their permuted colors.�

4.4 Congruence properties in 3-colorable graphsA properly 3-colored graph H is critical with respect to non-adjacent congruentvertices s and t in H , if for removed edge e from the graph H a proper 3-coloringof the graph H \e does not confirm the congruence of s and t vertices.

Property 4.4.1 Assume s and t non-adjacent vertices are congruent and lay ona cycle C (C∈B(H )) or on the unbounded face of a 3-colorable graph H . If H iscongruence critical with respect to (s , t), then end vertices (ξe, µe) of a removededge e from the graph H have the same color in a 3-colored graph H∪st\e.Proof . Suppose vertices ξe and µe have different colors, then H∪st would be 3-colorable which contradicts the assumption that the s and t vertices are congruent.�Lemma 4.4.2 There exists two 2-color paths between two congruent vertices.Proof . 3 In a coloring step, the congruent vertices have received the c3 colorand the non-colored enabling vertices receive colors {c1, c2}. The congruent verticescan only be initiated by two vertices of two triangles sharing an edge (Fig. 5.A).Any 2-color {c3, c} (c ∈ {c1, c2}) path from one congruent vertex to the otherexists. A new congruent vertex belongs to a triangle whose other two vertices areenabling and are adjacent to either one or to two congruent vertices. Thus exist two

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2-colors paths from the new congruent vertex to one, or to two congruent vertices.By recursive assumption, any two congruent vertices have two 2-color paths, sodoes the new congruent vertex. Besides, the length of a 2-color path between two congruence

pathcongruent vertices is even.� We will refer to such a path as a congruence path.�4.4.3 Congruence path analysisThe congruence path traverses through one or more triangulas. A path of enablingvertices is associated with one triangula, and it can be associated with two if anyenabling vertex of the first triangula is adjacent to an enabling vertex of the secondone. If a congruent vertex can be derived from two different sets of enabling verticesthen all the enabling vertices are relevant. Two enabling paths are independent ifvertices of one path are not adjacent to vertices of the other path. Note that a pathof enabling vertices is limited by adjacent to them congruent vertices.

Having only one path of enabling vertices and switching their colors does notproduce new paths. Assume there are two or more independent enabling paths.1 Let switch the {c1, c2} colors within a path of enabling vertices surrounded by thecongruent path of the {cC , c2} colors. The new path may go through the internalvertex of the c1 color switched to c2.2 Two independent paths of enabling vertices are limited by identical congruentvertices. Switching the colors in one path is equivalent to the alternative congruencepath of the other path of enabling vertices.

4.5 Coloring conflictsTheorem 4.3.1 includes the algorithm to 3-color a 3-colorable graph. Upon a col-lision, an expanded 4.3.1 algorithm evaluates the congruent vertices and detects acolor conflict of s and t vertices. Let a non-3-colorable sub-graph H conflict of H be H conflict

induced by the already colored vertices, whereas the graph H 3Clr = H conflict \ st is H 3Clr3-colorable. There exist H conflict graphs whose conflicts originate by applying therules 4.2.E in one step or in multiple steps.

4.5.1 One Step ConflictsA color conflict caused by congruent vertices received during one-step executionoccurs in two scenarios: 1. Congruent vertices are adjacent, and 2. Congruentvertices coerce an odd-length circuit.1 Adjacent vertices are assigned identical colors . The assigned color c3 to vertexs collides with the already colored vertex t . The enabling vertices with not yetassigned colors (OneSlot case of the 4.2.E2 and 4.2.E3 rules) receive {c1, c2} colors.The conflict sub-graph H conflict is limited to congruence path and its enabling ver-tices between s and t ; H conflict \ st is critical with respect to s and t . It is importantto point out that the congruence path is initiated by a unary congruence.2 Congruent vertices coerce non-3-colorable circuit . All vertices of an odd-lengthcircuit are adjacent to congruent vertices assigned in one coloring step. This circuitends up in a conflict regardless of how the colors are assigned to its vertices. Letus position the conflict color in vertices s and t adjacent to two directly congruentvertices. The conflict graph H conflict is induced by the circuit and congruent verticesenforcing this circuit, and by other congruent vertices and their enabling verticesthat are within the paths joining the enforcing congruent vertices. In contrast tocase 1, the sub-graph H conflict may not be critical.

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4.5.2 Multiple Steps Conflict - By ExhaustionWe propose the term “Conflict by exhaustion” for this scenario. When trying 3-color the graph H , all the choice vertices have exhausted their alternate colors andback-tracking reaches the initial triangle. Let the graph H conflict be induced by allthe colored vertices of the last coloring. Suppose the conflicting vertices s and t arenot in a triangle. The last colored triangle T is definitely part of the critical sub-graph of H conflict, and by Property 4.4.1 we can transfer the conflict to be betweens and t of T. The omitted edge st admits a 3-colorable graph H conflict\st .

Figure 7: 3-color conflictsIn the graph of Fig. 7.A congruent vertices s and t are adjacent (4.5.F1). Anyvertex in a triangle to which vertex t or s belongs to, can receive the color 3 byapplying different sets of enabling vertices. Congruent vertices 1 in the graphs ofFigs. 7.B & 7.C coerce odd length circuits (dashed lines - 4.5.F2). The graph in7.C has a restored non-significant vertex r . The graph in 7.D is a copy of Fig.3 in [11] with inserted cF vertices into faces of four or more edges (square boxesand dotted lines). The graph follows fixed color selection. Vertices u and v receivedifferent colors while they are congruent in a 3-colorable graph (Fig. 1 in [11]). Thisexample shows importance of a graph being 3-colorable when analyzing congruenceof its vertices. The graph in 7.E (Fig. 3 in [1]) illustrates a conflict “By exhaustion”.

4.5.3 Color conflictA color conflict is a non-correctable color collision of two adjacent vertices s andcolor

conflict t of the same color within the graph H conflict, whereas the graph H conflict \ st is3-colorable. We identify the following reasons for a color conflict.

4.5.F1 Two adjacent and congruent vertices are assigned the same color in acoloring step.

4.5.F2 Non-colored vertices of an odd-length circuit are adjacent to congruentvertices assigned the same color in one coloring step.

4.5.F3 The coloring exhausts the alternate colors for choice vertices.

Proposition 4.5.4 Graph is non-3-colorable if one of 4.5.F scenarios is confirmed.Proof . In a 3-colorable graph none of the 4.5.F scenarios occurs. The color de-pendencies are transparent for conflicts defined by 4.5.F1 or by 4.5.F2 scenarios, asdiscussed in 4.5.1. In one step, the rules 4.2.E1 and 4.2.E2 uniquely determine thecolor dependencies in a triangula, or in a neighboring triangula of at most distanceone (Lemma 4.2.2, OneSlot and ThreeSlots cases).

The third scenario does not indicate vertex colors dependencies leading to a con-flict. Each attempt to 3-color the graph may result in a different pair of conflictingvertices and Property 4.4.1 provides an explanation.

The three 4.5.F scenarios exhaust all the reasons for an unavoidable conflict,which, after all, needs two adjacent vertices to receive the same color. �

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5 Fixed Color Selection

Assume the sub-graphs {Gyy≺x} in Gx (Gx∈GkD, k≥2) have been properly 4-colored 2.2.2

and the vertices in the sub-rims {Ryy≺x} use three colors h+ = {c1, c2, c3}. When 3.1

expanding sub-graphs {Gyy≺x} to the tier T x, one of the h+ colors is designatedas the fixed color cF , not to be used by the rim Rx vertices. Fixed color selection 3.1.3

ensures the graph G(T x) with omitted cF color vertices is 3-colorable.

5.1 Auxiliary sub-graphs H 3Tx and H 3Gx of graph Gx

Assume the selected fixed color cF has been assigned to I x vertices. 2.2.3

Let H 3Tx be obtained from G(T x) by removing the cF color vertices. H 3Tx

Let H 3Gx be a sub-graph of Gx with vertices of the cF color removed. H 3Gx

5.2 Fixed color selection analysis in graphs GkD(k ≥ 2)

The analysis of paths {Px,yy≺x} between the rimRx and the sub-rims {Ryy≺x} allowsto select a subset of the same color vertices to become the designated cF color. Theselection ensures no congruence path exists between two adjacent vertices withinH 3Tx. Let’s remind that H 3Tx does not include the arcs of a Polygona part of asub-tier T y (y≺x ); potential color conflicts are addressed in Section 6.

Lemma 5.2.1 The sub-graph H 3Tx of Gx (Gx∈GkD, k ≥ 2) is 3-colorable.

Proof . We show that fixed color selection prevents two adjacent vertices frombeing congruent. There are two cases.A T x contains one part - Shell Sy.If a path Px,y (Y = {y}) is open, then its end vertices are separated at least byone vertex in Rx and by another vertex in Ry. Besides, a semi-closed path with thecF color assigned to a pivot prevents a conflict from occurring. A fixed color pivotwith “up()>2” changes the closed path into an open, and takes precedence over apivot with “up()=2” which changes the path into a semi-closed one.

Let the following symbols be used for specifying the subsequent mutually exclu-sive sub-cases A1, A2 and A3.

px,y = |Pivx,y| number of pivotsnx = |V(Rx)|, ny = |V(Ry)| number of vertices in Rx and Ry rimsmx,y = min{nx, ny} minimum of number of vertices in both rims

A1 px,y < mx,y ∨ (px,y = mx,y ∧ nx > ny)Exists a pivot p that one of the scenarios occurs.

1. up(p) > 22. Exists a cycle C (C∈T x) that |V(C )| > 3 and p∈V(C ).

A2 px,y = mx,y ∧ nx = ny = 3Any pivot blocks congruences in T x.

A3 px,y = mx,y ∧ (nx, ny > 3 ∨ (nx = 3 ∧ nx < ny))One of the following scenarios occurs.

1 Exists a pivot p that one of the scenarios A1.1 or A1.2 occurs.2 Exist pivots p and q that pq /∈Ry.3 Exists a pivot p with up(p) = {q , r} and exists a vertex v that belongs to Ry

and both p and v belong to a subset of the same color (Fig. 8.G). Vertex v blocks apotential congruence path between between q and r . Vertices p and v make H 3Tx

3-colorable when assigned the cF color. A congruence path between q and r existsonly if there are two additional pivots p1 and p2 adjacent to q and r repectively,that are enabling vertices for the path. If pivots p, p1, and p2 differ in colors thenbetween p1 and p2 must exist an enabling vertex v of the same color as pivot p.

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Figure 8: Fixed color selection - one Shell partThe selected fixed vertices use square boxes. The diagrams 8.A and 8.B showgraphs with open and semi-closed Px,y paths. The paths Px,y are closed in theFigs. 8.C. . . 8.F and illustrate corresponding cases: A1.1, A2, A3.2, and A3.3. Thegraph in Fig. 8.G shows a conflict in H 3Tx when the cF color is assigned to thewrong sub-set (case A3.3 fails); dark circles represent a congruence path.

B T x contains multiple parts .We assume the tier T x contains at least two Shell parts. A Tweel TRim or Polygonaparts do not contribute to a potential conflict within H 3Tx vertices. Besides, Shellswhich are not part of a MY can be analyzed independently, and then their non-cFcolors, if required, are synchronized according to Property 2,5.C2.

Further analysis focuses on linked parts of TShl, Sy, and MY types. The internalvertices of TShl and all Shell parts have initially been assigned the cF color. Weuse a tree traversal (Property 2.5.C3) within parts to synchronize the cF color, andbegin with accepting the fixed color of an existing TShl or any Shell part.

The fixed color re-selection in the next part is required when vertices of the cFcolor in both parts are adjacent. Each type of a new part has the following solution.1 - TShl. Assign a non-cF color to the internal vertex of a TShl part.2 - Sy. An alternative sub-set of the sub-rim vertices in the new part is chosen.There are scenarios that a conflict may occur for consecutive synchronized Shells.2.1. Two Shells share a vertex and |V(Rx)| = 3 (Fig. 9.A).2.2. A link is an enabling edge of congruent and adjacent vertices(Fig. 9.B).2.3. Congruent vertices of one Shell enforce an odd-length circuit (Fig. 9.B).

The solution applicable to all cases is to choose an alternative sub-set for onepart that contains an enabling vertex of the congruence path; the other part hasthe cF color synchronized as required. The chosen sub-set receives the fixed color,breaks the congruence path and avoids a conflict (Figs. 9.A and 9.B). �

Figure 9: Fixed color selection - color synchronization in multiple partsThe graph in Figs. 9.A shows color adjustment of two Shells sharing a vertex.The graph in Fig. 9.B presents initial color synchronization of individual parts insequence indicated by numbers in italics (2.5.C3). Two corrections are necessary:2.2 - a link between shells anables a congruence path, 2.3 - congruent vertices ’1 ’ areadjacent to a triangle {r , s , t} vertices. Small circles represent color adjustments.

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6 G2D Graphs

This section focuses on 4-colorability of G2D graphs. We begin with the Fritsch 2.2.2

graph ([8] p. 176) whose fixed color selection fails to receive a 3-colorable H 3Gx. 5.1

6.1 Class F of graphsA graph F is in the class F if F∈G2

D, F contains a Polygona-based leaf tier, and F

includes an arc-triangle cycle Tarc that consists only of arcs. Tarc

The class takes its symbol F from the Fritsch graph F (Fig. 10.B) which containsa Tarc. Regardless of which color is selected as the fixed one, the sub-graph H 3Gx

of F is not 3-colorable. On the other hand, some sub-graphs H 3Gx derived by fixedcolor selection can be 3-colorable (for example the graph F with removed any E x

edge). We define the class F for two reasons.1 It is simple to recognize its members.2 There exists an associated 3-coloring procedure (Lemma 6.1.2).

Figure 10: Examples of F graphsThe graph in Fig. 10.A shows the coloring of the F ’s leaf tier. The graph in Fig.10.B presents F and its H 3Tx 3-coloring, where Tarc arcs (dotted lines) are omittedfrom the analysis; the graph H 3Gx is not 3-colorable. The graphs in Figs. 10.C(D)are other examples for which fixed color selection fails to derive a 3-colorable H 3Gx.

Lemma 6.1.2 Exists assignment of the fixed color to Ry vertices to make H 3Gx ofF (F∈F) 3-colorable.Proof. By Lemma 5.2.1 the sub-graph H 3Tx of F is 3-colorable and we want toexpand 3-colorability onto H 3Gx. The sub-graph H 3Tx hides adjacencies of the endvertices of arcs for which a congruence path may exist resulting in a conflict.

1 Assume exists one Tarc, V(Tarc) = {r , s , t} and let (V upr , V up

s , V upt ) = (up(r),

up(s), up(t)). If exists a subset, let say V upr , such that |Vr| > 1, then fixed color

selection suffices to 3-color H 3Gx. By choosing the fixed color of vertex r no con-gruence path exists between s and t . Note that the end vertices of a non-Tarc arcare either between cF vertices or the arc’s end vertices are adjacent to cF verticesthat prevent forming a congruence path between them (Theorem 6.3, case 2).

In further analysis we assume (V upr , V up

s , V upt ) are single vertices (rup, sup, tup).

Suppose (rup, sup, tup) are unique. There is a pivot between two Tarc vertices andwhen the pivots are assigned the cF color, no congruence path exists between twoTarc vertices (Fig. 11.A). Suppose then that two or all of (rup, sup, tup) vertices areidentical. We apply fixed color selection; one of Rx vertices, let say r , is assignedthe cF color. Between s and t exists in the rim Ry a vertex that is also assigned thecF color. This vertex and vertex r block a congruence path from s to t (Fig. 11.B).

2 There are two or more Tarcs. Let T∗arc be such Tarc that two of its arcs betweentheir end vertices have no vertex that belongs to another Tarc, and (r

∗up, s∗up, t

∗up)are unique. If such a T∗arc does not exist, then fixed color selection ensures H 3Gx is3-colorable. Let these two arcs share a vertex r and let v (v∈ Ry, v 6=r) be adjacent

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to both vertices s and t (Fig. 11.C&D). Let a graph G∗arc be induced by the vertices:V(T∗arc), v , up(V(T∗arc)), and the remaining vertices between s and t (Fig. 11.D).For all G∗arc in H 3Tx we assign the cF color to the pivots between sr and rt vertices.For each G∗arc we test whether a collision exists within G∗arc. Consider sub-cases.

2.1 Collision exists . We assign the color cF to vertex v , making G∗arc 3-colorable.The reason is that vertex v disables a congruence path between two rimRx adjacentvertices, where r and s are enabling and t is a congruent vertex (Fig. 11.D).

Vertex v may be adjacent to a pivot p of neighboring Tarc or T∗arc with alreadyassigned the cF color. In this case v is one of the arc-triangle vertices and makesthe induced graph Garc or G∗arc 3-colorable by disabling a congurence path.

2.2 There is no collision. If vertex v is adjacent to a vertex of the cF color, thenthe next T∗arc is analyzed as described above.

Regardless of the sub-case, we follow Fisk’s 3-color distribution beginning with3.2

the colors of vertices s , t , and v . The distribution terminates when all H 3Gx verticeshave been assigned colors, or another T∗arc occurs with no shared up() vertices withthe current T∗arc; that leads to repeated analysis of the cases 2.1 or 2.2. In theiterative process, sub-graph H 3Gx becomes 3-colorable. �

Figure 11: Coloring G2D - Polygona with Tarc

The graph in Fig. 11.A illustrates fixed color assignment to pivots. The graphin Fig. 11.B presents fixed color selection. The graphs in Figs. 11.C & D showmultiple Tarcs; the vertics of a G∗arc graph are marked by dashed lines.

Lemma 6.2 Sub-graph H 3Gx of Gx (Gx∈GkD \ F, k ≥ 2) is 3-colorable when |Rx|

> 3 and all sub-graphs {Gyy≺x} have been 4-colored.Proof . The rim Rx vertices do not form a triangle and no pivots are congruentto them. Suppose non-adjacent Ry vertices p and q are congruent in Gy. Whenfollowing fixed color selection, a potential conflict between p and adjacent to itvertex v congruent to q in H 3Tx is blocked by the cF vertices between p and q . �

Theorem 6.3 Graphs G2D are 4-colorable.

Proof . Assume a graph G (G∈G2D) is two levels deep. By Lemma 5.2.1, the sub-

graph H 3Tx is 3-colorable. Consider a Shell encircling one of the following parts.1 Tweel TRim. The maximum number of fixed color vertices in the rim equalsb|V(Ry)| ÷ 2c and the fixed color is placed in almost every second rim’s vertex.

Suppose |V(Rx)| equals three. By Lemma 5.2.1, one pivot assigned the cFcolor suffices to have H 3Tx 3-colorable. However, a conflict occurs when |V(Ry)| isodd and exists vertex v (v∈V(Rx), down(v) ≥ 2), where two adjacent Ry vertices4.2.3

enable congruence to the Tweel’s internal vertex s , and the pivot t of not cF color iscongruent to v (Fig. 12.A). This scenario can be located and the conflict is avoidedby exchanging the fixed color with one of vertices enabling the congruent vertex s(Fig. 12.B). By Lemma 6.2 graphs with |Rx| > 3 and with no Tarc are 3-colorable.

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2 Polygona - no arc-triangle. We assume a Polygona’s rim Ry contains at leastfour vertices and no Tarc exists. The designated cF color selection follows Lemma5.2.1. If |V(Rx)| > 3 then criteria A1.2 is used. Otherwise when |V(Rx)| = 3 thencriteria A2.1 and A1.2 are applied.

Let analyze the end vertices of Polygona’s arcs, not discussed in Lemma 5.2.1.Any two rim’s Ry non-cF adjacent vertices are adjacent to the same cF vertex.This fact makes impossible to have a congruence passing through the Polygona.Now suppose vertex s is one of the st arc vertices. Vertex s is either between twovertices of the cF color, or both s and t vertices are adjacent to vertices of the cFcolor, that is, along the sub-rim between s and t there is always a vertex of the cFcolor. In either case, the neighboring cF vertices guarantee that s and t are notcongruent in the graph H 3Gx.

3 Polygona with arc-triangles . Lemma 6.1.2 proves H 3Gx 3-colorability.

4 Shells , Tweels (TRim, TShl), Polygonas . By Lemma 5.2.1 H 3Tx is 3-colorableand we want to expand 3-colorability onto H 3Gx. The colors of rim’s Rx verticesshared by two parts are synchronized that share two rim’s Rx vertices There aretwo potential corrections. 1. A rim Rx vertex shared by two neighboring Shells inMY is not congruent to an adjacent sub-rim vertex because the sub-rim has alreadybeen analyzed and assigned the cF color. The colors of one part can be synchronizedfor neighboring part, that is a part sharing two rim Rx vertices (Property 2.5.C2).

Re-assuming, the graph G is 4-colorable and its border is 3-colorable. �

Figure 12: Coloring G2D graphs

The graph in Fig. 12.A contains the maximal number of pivots assigned the fixedcolor, an example of a conflict discussed in TRim (case 1) and corrected in Fig.12.B. Fig. 12.C illustrates Polygona coloring (case 2); vertices of the dashed arcsuse different non-cF colors. Fig. 12.D shows a synchronization sequence (1 , . . . , 5 )of parts where color modified vertices use smaller shapes.

7 Fixed Color Refinement - Conflict Resolution

By Lemma 6.2.1, fixed color selection ensures no conflict exists for |V(Rx)| > 3and we assume |V(Rx)| = 3. Fixed color selection may not prevent a conflict in 5.1

3-coloring H 3Gx or a conflict may occur when re-inserting a non-significant vertex in 2.2.2

GkD graphs (k ≥ 2). Fixed color refinement modifies fixed color locations in H conflict 3.1.1

in order for H 3Gx to become 3-colorable.Let h+ = {cC , c1, c2}, cF = Π\h+ where cC is the conflict color of s and t vertices 3.1

in H conflict graph whereas H 3Clr = H conflict\st is 3-colorable. 4.5.3

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7.1 Refinement contourA refinement contour is a 2-colored odd-length closed path Cnt(c1, cC) in thecontour

graph H conflict where Cnt(c1, cC) = (v 1, v 2, . . . , v 2l+1) l ≥ 1; v 1 = s , v 2l+1 = t ,and c(v 2i+1) = cC (i = 0, . . . , l), c(v 2i) = c1 (i = 1, . . . , l).The contour satisfies the following conditions.7.1.G1 Only the colored vertices necessary to receive a conflict are included.7.1.G2 The contour includes at least one internal vertex of the c2 color which

cannot be modified to c1 or cC without a collision with a contour vertex.The contour may encircle cF vertices.

7.1.G3 Does not exist an odd-length circuit coerced by non-internal congruentvertices, where circuit is formed by vertex t and related to contour verticesof colors: non-internal c1, internal cF adjacent to c2, and external c2.

By 7.1.G2, if no c2 vertex exists to receive cF color then no exchange of colors ispossible; vertex c2 is necessary to break vertex t color congruences. The condition7.1.G3 ensures a refined graph H conflict becomes 3-colorable; no odd-length circuitof two colors is coerced by congruent vertices in the refined graph.

Figure 13: Refinement contours - invalid initial and valid alternativesIn Figs. 13.A. . . 13.E contours use thick lines, edges between enabling vertices aredrawn with dashed lines, non-colored vertices keep 0. Initial contours in Fig. 13.Afail: Cnt(3,2) - 7.1.G1 and not marked Cnt(3,1) - 7.1.G2. Fig. 13.B shows a validcontour with switched colors of enabling vertices. In Fig. 13.C Cnt(3,1) fails 7.1.G1and in Fig. 13.D Cnt(3,2) fails 7.1.G3 (fixed refinement would contain an odd-length circuit of vertices marked by thick lines). Conflicting vertices {s , t} in Fig.13.C are changed to {s∗, t∗} in Fig. 13.D; Fig. 13.E contains a valid contour.

Lemma 7.2 A graph H conflict contains a refinement contour.Proof . A refinement contour cannot be entirely embedded within H 3Tx (Lemma5.2.1) or in H 3Gy (by assumption H 3Gy is 3-colored). Let us locate a two-color path{cC , c1} between the conflicting vertices of s and t in the 3-colorable graph H 3Clr.By Lemma 4.3.4 such a path exists and the contour derivation cases correspond tothe originating conflict scenarios 4.5.F1, 4.5.F2, and 4.5.F3.1 Congruence direct . Within H 3Clr, exist vertices a and b initiating a congruencepath (a∈V(Rx), b∈V(H 3Gy) where y≺x ). Very often b is a non-cF pivot in V(Ry)when |V(Rx)| = 3. We require all vertex t enabling vertices be within the contour.1.1 Unary congruences . Note that a simple congruence uses only one pair of en-abling vertices. If no enabling vertex of t is inside the contour, then we switchthe t ’s enabling colors. Beginning with vertex t , a 2-color (c1, c2) path reaches theenabling vertices of the initiating simple congruence. This path is open because theinitating enabling vertices are adajcent only to the congruent vertices.

1.2 Unary and binary congruences . Suppose the condition 7.1.G1 is not satisfiedwhen not observing the execution of 4.2.E rules. In this case, vertex t belongs to

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a triangle T = {t , . . . } enclosed by the initial contour (Fig. 13.C). In contrast tounary congruences, each T vertex can become a congruent one (Fig. 13.D) and analternative contour of different congruent vertices would be derived (Fig. 13.E).

Suppose the contour fails condition 7.1.G3 which means the refined conflictgraph contains an odd-length circuit C induced by the contour external congruentvertices. There exists one or more enabling vertices {r , . . . } that belong to C andare not inside the contour (Fig. 13.D). By 4.4.3, a part of the contour may have itsenabling colors switched to include some of the {r , . . . } vertices, ensuring 7.1.G3is satisfied and keeping the t ’s enabling vertices inside the contour (7.1.G2). If allenabling vertices are in one path, then the intersection of graphs induced by bothcontours (Fig. 14.B) includes vertices of the c2 (c1) color that belong to the firstcontour and are inside the second contour (or vice versa). Hence, if one contour isinvalid then the other contour is valid because the odd-length circuit is broken.

Figure 14: Refinement contours - unary and binary congruencesEdges between enabling vertices are drawn with dashed lines. The graph in Fig.14.A shows two independent paths of enabling vertices. In the graph of Fig. 14.Bthere exists only one path; vertices that belong to the intersection of two contour-graphs are marked by thick lines.

Figure 15: Refinement contours - coerced and “by exhaustion”In Figs. 15.B, 15.C contours are formed by combining paths (thick lines) of con-gruent vertices (1 , 3 ) and odd-length circuits (dashed lines). Graphs in 15.A, 15.B,15.C are the same as in 7.A, 7.B, and 7.C. The graph in 15.D shows “By exhaustion”conflict with cF vertices shown as square boxes, and their edges (dashed lines).

2 Congruence coerced . A conflict occurs when congruent vertices of c1 color coercean odd-length circuit C (including triangle) of two colors {c3, c2}(cC = c3). Thecircuit C can be recolored in such a way that the conflicting vertices (s , t) of colorcC are adjacent to two congruent (p, q) vertices. By Lemma 4.3.4 exists a two-color{c1, c3} path P between p and q . The color cC is common to the path P and thecircuit C vertices. We form a contour by adding the (s , t) vertices to the pathP . The circuit C can be enclosed by the contour (Fig. 15.B) or it can be locatedoutside the contour (Fig. 15.C). Consider two cases with respect to C location.2.1 Inside contour . The lengths of internal {c1, cF} and external {c1,c2} 2-colorpaths are even and so is their combined path. No internal 2-color path exists fromvertex t to a contour c1 vertex, blocked by the circuit C internal vertices.

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2.2 Outside contour . An external 2-color {c2, c1} path from t to a c1 contour vertexq is blocked by vertices cC of P and by contour internal and enabling vertex of thec2 color, while an internal path from t to q may exist.3 By exhaustion. The graph Gconflict includes a triangle to which the conflictvertices s and t can be transferred (4.5.2), although s and t are not derived by thecoloring rules 4.2.E. The contour encircles this triangle and its third vertex is notan enabling one but satisfies 7.1.G2. Vertices s and t also belong to another cycle([6] p.85)), which is contour-external. Because the distance between triangles is atleast one, then this cycle contains at least four vertices, of which vertex t is notadjacent to a c2 vertex. �

7.3 Fixed color refinementFixed color refinement changes the enclosed contour vertices of the color c2 tocF , whereas adjacent vertices with the color cF are assigned a color in Π \ cF .

Lemma 7.4 Fixed color refinement resolves conflicts in the graph H 3Gx.Proof . Assume that adjacent vertices s and t have been assigned the conflict cCcolor in H 3Clr while 3-coloring the graph H 3Gx. By Lemma 7.2, a contour of colors{cC , c1} can be located in H conflict. The refinement makes H conflict 3-colorable bybreaking derivability of the congruent vertices inside the contour and by ensuringno odd-length circuit of two colors is formed within the modified H conflict graph.Consider the following familiar cases.1 Congruence direct . Vertex t is assigned the color c∈{c1, c2} and vertices do notform an odd-length circuit.

2 Congruence coerced . Vertices s and t are not congruent and one of them can beused to initiate re-coloring. The color assigned to t depends on the circuit location.2.1 Inside contour . Vertex t is assigned the c2 color and refining does not generatean odd-length circuit of the {c1, c2} colors. The c1 color of contour’s congruentvertices is preserved2.2 Outside contour . Vertex t gets the c1 color and breaks an odd-length 2-colorcircuit C . Contour’s vertex adjacent to t receives the c2 color and no external pathof the {c1, c2} colors exists, while an internal path of the {c1, c2} colors may exist.

3 By exhaustion. The exchange of the colors {cC ,c2} begins with assigning the colorc2 to vertex t which is not adjacent to a vertex of color c2 outside the contour. �

Figure 16: Fixed color refinementThe graphs in Figs. 16.A, . . . , 16.D illustrate fixed color refinement for the contourgraphs in Figs. 14.A, . . . , 14.D. Graphs that provide counter-examples to Kempe’sproof [9](for example, the Errera, Heawood, . . . graphs) do not require refinement.

Proposition 7.5 Graphs G∈GkD (k≥1) with absent non-significant vertices are

4-colorable.Proof . By Theorems 3.2.1 and 6.3, graphs of one and of two levels deep are 4-colorable. By Lemma 7.4, the graph H 3Gx (H 3Gx ⊂ Gx, G = Gx) is 3-colorable.Hence, G is 4-colorable, where G is a graph of any depth k . �

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8 Restoring Non-significant Vertices

The re-inserted non-significant vertices preserve the graph’s 4-colorability as shownin Lemma 8.1.2. The Four-Color Theorem concludes the paper.

8.1 Re-insertion of non-significant verticesBy Lemma 7.4 and Proposition 7.5, a graph G (G∈Gk

D, k ≥ 1) without non- 2.2.2

significant vertices is 4-colorable. Assume hroot+ = {c1, c2, c3} are the colors of 3.1

the root rim Rroot and the fixed color cF is Π\hroot+. Let’s recall that only rootrim vertices do not use the cF color (3.1.D2). The re-insertion of the non-significantvertices could result in the hroot

+ colors modifications of some of the graph’s vertices,and a required fixed color refinement would even modify the cF color placement.

Let I rule contain any internal contracted vertex and the re-inserted non-significantvertices for the rule, V rule - the rims vertices adjacent to I rule vertices, K rule - the 2.3

colors of V rule vertices (K rule = K(V rule)).Suppose |K rule| ≤ 2. Then two available colors {Π\K rule} suffice to color the

internal vertices I rule, including the I rule vertices for the rule 2.3.B1.We assume |K rule| = 3 in further analysis. Given below algorithm verifies

whether the re-inserted non-significant vertices I rule can be assigned colors withoutmodifying the colors of encircling them V rule vertices.

Algorithm 8.1.13 Assume V rule vertices are colored and are adjacent to not necessarily all I rule

vertices. Initially vertices I rule are non-processed (available colors are not assigned).In each iteration select an unprocessed vertex r (r∈I rule) that is either adjacent

to a processed vertex p (p∈I rule), or is adjacent to a colored vertex q (q∈V rule).The colors available to vertex r are derived by the formula.

Ar = Π \(K({p1, . . . }) ∪ {B q1 , . . . } )

where {p1, . . . } are V rule vertices adjacent to r , {q1, . . . } are processed I rule verticesadjacent to r and {B q1 , . . . } are calculated as follows:

B qi = if (|Aqi | ≥ 2) then ∅ else Aqi for i = 1, . . .If Ar results in one color, then each processed vertex q i adjacent to r has its

subset of available colors reduced to Aqi\Ar. The color reduction is propagatedfurther if the reduced set of available colors Aqi results in one color.

Iterations continue until all I rule vertices are assigned colors, or no color is avail-able to r . An empty set may also result during propagation the color reduction.

Assume available sets of colors Ar (|Ar| ≥ 1 for r∈I rule) are non-empty. Theyare transformed into a feasible coloring as follows. Vertex r whose Ar contains twoor more colors is assigned a color c in Ar, Ar is reset to c, and each vertex q∈I rule

adjacent to r has its Aq reset to Aq\c. The color reduction is propagated further ifthe reduced set Aq results in one color. This time color reduction in vertex r doesnot deprive an adjacent vertex q of the only color because in the first place vertexq would propagate reduction of this color to vertex r . The I rule vertices do notinduce a cycle and the reduction propagation always produces a correct result. �

Lemma 8.1.2 Restored non-significant vertices preserve graph’s 4-colorability.Proof . Beginning with the leaf tiers and moving recursively up to the root tier,we restore the graph’s G vertices and edges suppressed by the rules 2.3.B1 through2.3.B4. If |K rule| ≥ 3, then we use Algorithm 8.1.1 to assign colors to I rule vertices.Non-significant vertices contracted to a vertex hide adjacencies and the algorithmmay fail to assign a color to vertex r . We consider cases corresponding to the rules.

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A Rule 2 .3 .B4 . This rule applies to TRim part, which may appear in a not neces- 2.4.3

sarily leaf tier T y. Recall that initially two colors were intermittently assigned toleaf’s rim vertices, and a third color c was assigned as needed (Theorem 3.2.1). Wewill distinguish the following sub-cases.

A1 G∈G1D. The cause of the algorithm failure is the wrong selection of a vertex to

keep the third color. Let assign the third color c to r and begin a 2-color exchangepath {c, c2} along the rim (Fig. 17.A).

A2 G∈G2D. Let nx = |V(Rx)| (x = root). Restored vertices may cause a conflict

when nx = 3; surprisingly, there is no conflict when nx > 3. There are two causesof the algorithm failure: 1. The graph G3Gx is not 3-colorable, and 2. The graphG3Gx is 3-colorable but the third color is placed in the wrong vertex.

A2.1 Crucial is the scenario when two pivots are adjacent and nx = 3 (Fig. 17.B);one of these two pivots p keeps a non-fixed color and is congruent to a rim Rx vertexr . A conflict only occurs when p becomes congruent to a sub-rim Ry vertex whichis adjacent to r . The solution is to assign the fixed color to p (Fig. 17.C) whichblocks internal and tier T x congruences, making H 3Gx 3-colorable.

A2.2 We assume that nx > 3. By Lemma 6.2.1, no conflict exists (Fig. 17.D).

Figure 17: Re-insertion of non-significant vertices into leaf tierA G1

D graph in Fig. 17.A shows a conflict for restored vertices and a correction.Figs. 17.B and 17.C illustrate case A2.1. Cases A2.2 and A3.2 are shown in Figs.17.D and 17.E (congruent vertices 1 coerce a conflict).

A3 G∈GkD (k > 2). Both A2.1 and A2.2 sub-cases also apply to graphs of depth

greater than two. Let T y be a leaf tier and T x its sup-tier in G . If colors of threeRx vertices are modified (case A2.1), then a simple mapping of colors within Gx

would restore the original T x colors. The case A2.2 works for each sup-tier on itsway up to the root tier, provided each rim contains more than three vertices.

Otherwise a conflict may occur because of possible congruences of the rim’svertices with pivots (Fig. 16.E). A permissible 2-color path includes vertex r (notassigned yet color) and an adjacent vertex p with the c1 color. We require the colorc1 of p and assigned color c2 to r do not collide with the color of the third vertexq in an existing (r , p, q) triangle. The cF color is excluded from a 2-color path.There are two cases.

A3.1. A permissible path of two colors (c1, c2) does not exist from s to t . Toresolve the conflict, we assign the color c1 to vertex r and recolor the open path.

A3.2. Each permissible path between s and t vertices results in an odd lengthcircuit. If the conflict is caused by 4.5.F1 or 4.5.F2 scenario, then we will use therefinement to resolve the conflict. Otherwise, we 3-color the graph H 3Gx and applyfixed color refinement if needed.

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B. Rules 2.3.B2 and 2.3.B3 for GkD (k ≥ 2) graphs.

B1 Rule 2 .3 .B2 . A set of Vrule vertices include the contracted vertex v (v∈Ry) andadjacent to it Rx vertices (y≺x ) while Irule contains the re-inserted non-significantvertices. We assume that the coloring algorithm 8.1.1 has failed. By applying A3.1or A3.2 cases the conflict might be resolved. If not, we will use fixed color selection.

B2 Rule 2 .3 .B3 . Sub-rims vertices adjacent to an internal contracted vertex donot participate in this rule. The internal vertex still can be adjacent to multiplerim Rx vertices; this is the exact same scenario as the case of B1. �

Figure 18: Re-insertion of non-significant vertices between rimsRestored vertex r in 18.A causes a coerced conflict (4.5.F2). The graph of Fig.18.B is 4-colored, in Fig. 18.C non-significant vertices are inserted into a leaf tier(f - resolved) and a middle tier (r - requires fixed color refinement). The graph ofFig. 18.D shows the solution and includes vertex v inserted into the root tier; the2-color (cF , c2) exchange keeps graph 4-colorable. When vertex p receives the cF

color, then the exchange (c3, c2) also works.

Proposition 8.2 A planar graph is 4-colorableProof . Beginning with the leaf tiers and proceeding recursively up to the root tier,the entire graph without non-significant vertices is properly 4-colored as provenby Theorems, Lemmas, and Propositions in Sections three and five to seven. ByLemma 8.1.2, the re-inserted non-significant vertices preserve 4-colorability of theoriginal graph. This finalizes the algorithmic proof of the Four-Color Theorem.QED

AcknowledgementsI would like to thank Andrzej Proskurowski for his remarks on the initial pages of anearly version and I am grateful to the anonymous reviewers for their comments andsuggestions for improving the quality and readability of the paper. Many thanks toEdanz and PaperTrue editors for linguistic support.

This research did not receive any specific grant from funding agencies in thepublic, commercial, or not-for-profit sectors.

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