GRAPHSWITHODDCYCLE LENGTHS 5 AND 7ARE 3-COLORABLE*
TOMÁŠ KAISER†, ONDŘEJ RUCK݇, AND RISTE ŠKREKOVSKI§
Abstract. Let LðGÞ denote the set of all odd cycle lengths of a graph G. Gyárfás gave an upper boundfor χðGÞ depending on the size of this set: if jLðGÞj ¼ k ≥ 1, then χðGÞ ≤ 2kþ 1 unless some block of G is aK2kþ2, in which case χðGÞ ¼ 2kþ 2. This bound is generally tight, but when investigating LðGÞ of specialforms, better results can be obtained. Wang completely analyzed the case LðGÞ ¼ f3; 5g; Camacho provedthat if LðGÞ ¼ fk; kþ 2g, k ≥ 5, then χðGÞ ≤ 4. We show that LðGÞ ¼ f5; 7g implies χðGÞ ¼ 3.
1. Introduction. Let LðGÞ, or for short L, be the set of all odd cycle lengths of agraphG. One may investigate the relation between this set, or its size, and the chromaticnumber χðGÞ of G. For example, it is well known that jLðGÞj ¼ 0 precisely whenχðGÞ ≤ 2. The following result, originally conjectured in a weaker form by Bollobásand Erdős and later, in the version presented here, by Gallai, states that there is a gen-eral upper bound for χðGÞ in terms of jLðGÞj.
THEOREM 1.1 (Gyárfás [5, Corollary of Theorem 1]). If jLðGÞj ¼ k ≥ 1, then thechromatic number of G is at most 2kþ 1 unless some block of G is a K2kþ2. (If thereis such a block, then the chromatic number of G is 2kþ 2.)
This bound is globally tight for any k, as evidenced by K2kþ1. However, the follow-ing results demonstrate that it can be improved for particular cases. Wang [8] provedthat if LðGÞ ¼ f3; 5g, then χðGÞ ¼ 3 unless there is a K4 or a wheel on six vertices in G,in which case χðGÞ ¼ 4 unless G contains a K5. Camacho [1] showed that χðGÞ ≤ 4whenever LðGÞ ¼ fk; kþ 2g, k ≥ 5.
In this paper, we concentrate on a special class of graphs with L ¼ f5; 7g, the mainresult being the following theorem.
THEOREM 1.2. Every graph with L ¼ f5; 7g is 3-colorable.The theorem definitively refines Camacho’s result [1] for this case. Although the last
step of the proof is closely tailored to the particular class of graphs examined, the rest ofthe argument works for all graphs with L ¼ fk; kþ 2g, k ≥ 5.
The paper is organized as follows. Section 2 includes a few general, mutuallyunrelated lemmas used later. The purpose of section 3 is to state and proveTheorem 3.1—a slight strengthening of Theorem 1.1 for graphs with jLj ¼ 1 that isneeded in the subsequent argument. Finally, in section 4 we first restrict the structure
*Received by the editors June 12, 2009; accepted for publication (in revised form) September 3, 2010;published electronically July 19, 2011. This work was supported by the Czech-Slovenian bilateral researchproject 15/2006–2007, Research Plan MSM 4977751301 of the Czech Ministry of Education, and projectGAČR 201/09/0197 of the Czech Science Foundation.
http://www.siam.org/journals/sidma/25-3/76186.html†Department of Mathematics and Institute for Theoretical Computer Science, University ofWest Bohemia,
Univerzitní 8, 306 14 Plzeň, Czech Republic ([email protected]). Institute for Theoretical Computer Scienceis supported by the Czech Ministry of Education as project 1M0545.
‡Department of Mathematics, University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic([email protected]).
§Department of Mathematics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia ([email protected]). This author is supported in part by ARRS Research Program P1-0297.
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of 4-critical graphs with L ¼ fk; kþ 2g, k ≥ 5, and using that, we prove the main result,Theorem 1.2.
For the convenience of the reader, the text is accompanied by a number of figures, allof which obey the following rules of style. Vertices are shown as small filled discs. Solidlines, usually straight, stand for single edges; dashed lines, either curly or curved, areused for paths and cycles. Subgraphs that are not depicted completely by showing alltheir vertices and edges are typically represented by light or dark gray regions with thindotted lines as their boundary. Next, boldface is sometimes used to distinguish (impor-tant) objects. On the other hand, the possible nonexistence of a particular entity is al-ways indicated by dotting its standard representation in the case of an edge or a path or,for a vertex, by replacing the ordinary disc with a circle. Last, labels that are not namesof objects, i.e., those indicating lengths of paths or colors of vertices, are enclosed inbrackets.
In the remainder of this section, we mention several notions and results that areparticularly important for the next discussion or are not as well known. Besides these,we use only standard graph theory concepts and notation, which can be found inDiestel’s monograph [2], for example.
Due to the nature of the problem, we confine ourselves to simple graphs, that is,(undirected) graphs without multiple edges and loops. Let G be a graph with theset of vertices V and the set of edges E; we also write G ¼ ðV;EÞ and refer to Vand E as V ðGÞ and EðGÞ, respectively. Then by jGj we mean jV ðGÞj, that is, the num-ber of vertices ofG; kGk denotes jEðGÞj, the number of edges ofG. Hence, whenever P isa path, kPk is the length of P; if C is a cycle, its length is equal to jC j ¼ kCk. The graphG is called trivial if jGj ≤ 1. For the path P, an internal vertex is any of its vertices that isnot an end-vertex of P. Two paths P1 andP2 are internally disjoint if neither contains aninternal vertex of the other. A k-cycle is a cycle of length k. When two vertices x, y ∈V ðGÞ are at distance k in G (i.e., k is the length of a shortest path joining x and y in G),we write dGðx; yÞ ¼ k. Next, the set notation is extended in the following manner. LetG1
and G2 be graphs. By G1 ⊆ G2 we mean that G1 is a subgraph ofG2; G1 ∪ G2 and G1 ∩G2 denote the graphs ðV ðG1Þ ∪ V ðG2Þ; EðG1Þ ∪ EðG2ÞÞ and ðV ðG1Þ ∩ V ðG2Þ;EðG1Þ ∩ EðG2ÞÞ, respectively. If X is a set, G1 \ X stands for the graph with V ðG1Þ \X as its set of vertices and having all the edges of G1 not incident with any element of Xas its edges. We put G1 \ G2 ≔ G1 \ V ðG2Þ. For a vertex x, G1 \ fxg is written rather asG1 \ x unless there is a risk of confusion.
We use the notation of Diestel [2] for describing subpaths of paths or cycles andcombinations of these. If P is a path and x and y its vertices, then xPy denotes thesubpath of P between x and y including these two vertices. Moreover, if x or y is anend-vertex of P, we can optionally omit it, writing Py or xP, respectively. For the con-catenation of paths P1; P2; : : : ; subpaths x1R1x2; x2R2x3; : : : ; and single edgesxnxnþ1; xnþ1xnþ2; : : : ; sharing their end-vertices in sequence, such that the resultinggraph is a path or a cycle, we write, shortly, P1P2 · · · x1R1x2R2x3 · · ·xnxnþ1xnþ2 · · · . All these conventions are extended in a respective manner also for cy-cles, provided that the resulting description is unambiguous. For instance, to specify asingle subpath of a cycle C delimited by vertices x and y, we need to name a third vertexz lying on that path; therefore, we write xCzCy.
We adopt a few less common concepts used by, e.g., Diestel [2] and Voss [7]. LetG bea graph and H and I be its subgraphs. A nontrivial path in G is an H -path if it hasprecisely its end-vertices in common with H ; if this path is a single edge, then it is calleda chord of H . An H–I path is any (possibly trivial) path P ≔ u · · · v in G for which
1070 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
V ðP ∩ HÞ ¼ fug and V ðP ∩ I Þ ¼ fvg. If H or I consists of a single vertex x or y, re-spectively, we also write x–I , H–y, or x–y path instead of H–I path. By an H -bridge wemean any subgraph M of G which is either a chord of H or a component of G \ H to-gether with all the edges of G linking the component to the vertices of H . The verticesfrom V ðM ∩ HÞ are the attachments of M to H . It is clear that any vertex or edge of Gnot inH belongs to a uniqueH -bridge; that is,G is the union of all theH -bridges and thesubgraph H itself, and two different H -bridges intersect only in their common attach-ments. Other trivial properties ofH -bridges, often used later in the paper, we state in thesubsequent lemma.
LEMMA 1.3. Let G be a graph, H be its subgraph, and M be an H -bridge. Then(1) if M ≠ K2, no two of its attachments are adjacent in M ;(2) for every two vertices u and v of M , there exists a path P ≔ u · · · v ⊆ M such
that V ðP ∩ HÞ ⊆ fu; vg.Now we recall the following notion concerning graph coloring, which is crucial in
this paper. A graph G is k-critical for k ≥ 2 if it is not (k− 1)-colorable but each ofits proper subgraphs is (k− 1)-colorable. Clearly, G is then k-chromatic. For exam-ple, the 3-critical graphs are precisely the odd cycles. We will deal with 4-criticalgraphs, so we recollect some of their basic properties here. Directly from the defini-tion, every 4-critical graph G is 2-connected, and the minimum degree of G is atleast 3. One can easily see that G is even 3-edge-connected, which also follows fromTheorem 1 of Dirac [4] as a special case. However, simple examples show that G neednot be 3-connected. Dirac [3, statement (4), page 45] and Toft [6, Theorem 2.1] dis-cuss the presence of (k− 1)-critical subgraphs in k-critical graphs; we use part of theirresults formulated as the next lemma.
LEMMA 1.4. Let G be a 4-critical graph. Then for every two distinct vertices x and yof G, there is an odd cycle C ⊆ G \ x containing y.
Regarding the graph connectivity, we will often exploit Menger’s theorem in thefollowing form.
THEOREM 1.5. Let G be a k-connected graph, and let A and B be two sets of its ver-tices such that A ⊈ B and B ⊈ A. Then there are k distinct internally disjoint A–B pathsin G. If jAj ≥ k, then these paths can be chosen such that they have pairwise differentend-vertices in A. If jBj ≥ k as well, then there are k pairwise disjoint A–B paths in G.
We remark that the assumptions A ⊈ B and B ⊈ A are needed only for the casesjAj < k and jBj < k, respectively.
Finally, let us recall several facts and notions related to 2-connected graphs. A cut-vertex of a graph G is a vertex whose removal increases the number of components of G.A block of G is a maximal connected subgraph of G without a cut-vertex, that is, amaximal 2-connected subgraph, or a cut-edge, or an isolated vertex of G. Two blocksintersect in at most one vertex, which is then a cut-vertex of G. The block graph of G,denoted by BðGÞ, is the bipartite graph on the set of the cut-vertices of G and the set ofthe blocks ofG with xB being its edge if and only if x ∈ V ðBÞ. For any graphG, its blockgraph is a forest having no cut-vertex of G as its leaf. Further, it is connected if and onlyif G is connected.
2. Preliminary results. In this section, we accumulate several basic lemmas usedthroughout the rest of the paper. We start with a trivial statement.
LEMMA 2.1. Let x and y be two distinct vertices of a bipartite graph G. Then anyproper coloring of the subgraph of G induced by fx; yg can be extended to a proper3-coloring of G.
LEMMA 2.2. Let u, v, andw be three distinct vertices of a 2-connected graphG. Then
there exist a cycle C such that u, v ∈ V ðCÞ and a w–C path (possibly trivial) whose end-
vertex on C is different from both u and v.
Proof. Since G is 2-connected, there is a cycle C containing u and v. If w ∈ V ðCÞ,the assertion is true; otherwise, we can apply Theorem 1.5 to fwg and V ðCÞ, obtainingtwo internally disjoint w–C paths with distinct end-vertices w1 and w2 on C . If either ofthese vertices differs from both u and v, taking the respective path we are done. Theremaining case is that fw1; w2g ¼ fu; vg, but then the union of both the w–C paths andeither of the two subpaths of C delimited by u and v is a cycle going through all of u, v,and w—a desired configuration again. ▯
LEMMA 2.3. Let αi and βi, i ∈ Z3, be integers satisfying the following conditions:(1) αi þ αiþ1 equals βi þ βiþ1 or βiþ2, i ∈ Z3,(2) αi ≠ 0, i ∈ Z3, and(3) β0 þ β1 þ β2 is odd.
Then αi ¼ βi for all i.
Proof. Condition (1) yields a certain system of three equations for αi and βi. We
consider all its possible forms.Assume first that the number of the equations having their right side of the
form βi þ βiþ1 is even. Then by summing all the three equations, we obtain that 2ðα0 þα1 þ α2Þ equals β0 þ β1 þ β2 or, without loss of generality, β0 þ β1 þ 3β2; this is in bothcases impossible by condition (3). If exactly one of the equations has the term βi þ βiþ1
as the right side, one can infer that αiþ2 ¼ 0, a contradiction to assumption (2).The only remaining possibility is that the right sides of all the equations are in the
form βi þ βiþ1. Then by solving the system, we conclude αi ¼ βi for all i. ▯The next lemma, concerning odd cycles and paths joining them in 2-connected
graphs with L ⊆ fk; kþ 2g, L ≠ ∅, extends the results of Camacho [1, Lemma 3.1]and Gyárfás [5, Lemma 1].
LEMMA 2.4. Let G be a 2-connected graph with LðGÞ ⊆ fk; kþ 2g; let C0 and C 1 be
two odd cycles in G.(1) If jC 0j ≠ jC1j, then the cycles are not disjoint. If jC0j ¼ jC1j ¼ maxl∈LðGÞ l,
then jC 0 ∩ C1j ≥ 2.(2) Suppose that C0 and C1 are disjoint. Then
(a) LðGÞ ¼ fk; kþ 2g, and jC 0j ¼ jC 1j ¼ k;(b) every two disjoint C 0–C 1 paths P0 and P1 in G are both of length 1. Let xi
and yi, i ¼ 0, 1, be the end-vertex of Pi lying on C 0 and C 1, respectively.Then dC0
ðx0; x1Þ ¼ dC1ðy0; y1Þ;
(c) assume that there are three pairwise disjoint C 0–C 1 paths Pj, j ¼ 0, 1, 2, inG. If Pj
i , i ¼ 0, 1, denote the three subpaths of Ci delimited by theend-vertices of the paths Pj in such a way that V ðPj
i ∩ PjÞ ¼ ∅,then kPj
0k ¼ kPj1k;
(d) there are no four pairwise disjoint C 0–C 1 paths in G.For the situation of part (2c), see Figure 2.1(b).
Proof. For convenience, let ci, i ¼ 0, 1, denote the length of Ci. We start with a
general observation, whose notation is shown in Figure 2.1(a).
1072 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
Suppose that there is a pair of disjoint C 0–C 1 path P and Q such thatV ðC 0 ∩ C 1Þ ⊆ V ðP ∪ QÞ;let d ≔ kPk þ kQk. Next, let Ai and Bi, i ¼ 0, 1, bethe subpaths of Ci joining the end-verticles of P and Q, and let ai and bi referto their lengths. Then, without loss of generality, G contains two odd cycles oflengths lj, j ∈ Z2, for which
lj ¼ aj þ bjþ1 þ d; j ∈ Z2;ð2:1:1Þ
and
c0 þ c1 þ 2d ¼ l0 þ l1:ð2:1:2ÞWe prove the observation. As Ci is odd, ai and bi have different parity. Hence,
without loss of generality, lj ≔ aj þ bjþ1 þ d is odd for both j ∈ Z2, and since the pathsAj and Bjþ1 are internally disjoint, AjPBjþ1Q is a cycle of length lj. Next, summing theequations of (2.1.1) and using ci ¼ ai þ bi, i ¼ 0; 1, we obtain the formula (2.1.2).
Let us proceed to show statement (1). If jC 0 ∩ C 1j ≥ 2, there is nothing to prove. Sosuppose the contrary. As G is 2-connected, using Theorem 1.5 we can find paths P
and Q satisfying the assumptions of observation (2.1). Note that d > 0 by the conditionon jC 0 ∩ C1j; (2.1.2) thereby immediately excludes the possibility that c0 ¼ c1 ¼maxl∈LðGÞ l. Now let c0 ¼ k and c1 ¼ kþ 2 (or vice versa). Then, as the suml0 þ l1 is at most 2kþ 4, the formula (2.1.2) implies that d ≤ 1. Hence, at least oneof the paths P or Q is trivial; C 0 and C1 are not disjoint.
We continue with part (2). Statement (2a) follows directly from part (1); we focuson statement (2b). As V ðC 0 ∩ C1Þ ¼ ∅, we can put the paths P0 and P1 in place of Pand Q in observation (2.1). By assumption, d ≥ 2. Then (2.1.2) is satisfied only if d ¼ 2and lj ¼ kþ 2, j ∈ Z2. Hence, both P0 and P1 are of length 1 and, further, the formulasof (2.1.1) turn into the equations aj þ bjþ1 ¼ k, j ∈ Z2. As ai þ bi ¼ ci ¼ k, i ¼ 0, 1, bypart (2a), we easily derive the wanted equalities a0 ¼ a1 and b0 ¼ b1.
Now suppose that the assumptions of statement (2c) hold; see Figure 2.1(b). Forbrevity, let αj and βj denote kPj
0k and kPj1k, respectively. Applying part (2b) to all the
three pairs of paths which can be chosen from fP0; P1; P2g, we obtain that αk þ αkþ1
equals either βk þ βkþ1 or βkþ2 for all k ∈ Z3. As also αj ≠ 0 for all j and β0 þ β1 þ β2 isodd, Lemma 2.3 gives the desired conclusion.
We finish the proof by considering part (2d); the subsequent notation is shown inFigure 2.1(c). Assume to the contrary that there are four pairwise disjoint C 0–C 1 pathsin G. We adopt all the notation of statement (2c) for the first three of them; let P3
denote the fourth path. Next, let xki , i ¼ 0, 1, k ¼ 2, 3, be the end-vertex of Pk lyingonCi. Without loss of generality, x30 belongs to P
00; we refer to the subpath of P0
0 betweenx30 and an end-vertex of P1 as P3
0. Similarly, let P31 be the part of C 1 between x31 and an
end-vertex of P1 such that it is disjoint from P0.Now we apply part (2c) twice, to the triples of paths fP0; P1; P2g and fP0; P1; P3g.
It yields
kPj0k ¼ kPj
1k; j ¼ 0; 1; 2;ð2:2Þin the former case, and
kP30k ¼ kP3
1kð2:3Þin the latter one. If x31 is contained in P1
1, then clearly kP01k < kP3
1k, but as kP00k > kP3
0kby definition, we have a contradiction to the preceding equalities. By symmetry, onecould argue in a similar way that also x31 ∈= V ðP2
1Þ. Thus, x31 is on P01, and hence,
kx20P00x
30k ¼ kx21P0
1x31k by (2.2) and (2.3). Further, all of the four paths Pi,
i ¼ 0; : : : ; 3, are single edges, as asserted by statement (2b) It follows that there is acycle P0P2
1P1P3
0P3x31P
01P
2P10 of length kþ 4 in G—a contradiction. ▯
Finally, we include an obvious condition on the lengths of C -paths, C being an oddcycle, in graphs with L ⊆ fk; kþ 2g, L ≠ ∅.
LEMMA 2.5. Let C be an odd cycle in a graph G; let P ≔ u · · · v be a C-path. Then(1) if jLðGÞj ¼ 1, then dC ðu; vÞ ¼ minfkPk; jC j− kPkg. Thus, P is not a chord
of C ;(2) if LðGÞ ¼ fk; kþ 2g and jC j ¼ kþ 2, then dC ðu; vÞ equals either minfkPk;
kþ 2− kPkg or minfkPk þ 2; k− kPkg. That implies the following:(a) kPk ≤ kþ 1;(b) if P is a chord of C , then dC ðu; vÞ ¼ minf3; k− 1g.
Proof. We consider the two subpaths of C delimited by u and v. As the cycle C isodd, the length of one of these paths, say, P 0, must have parity distinct from that of kPk.Then P and P 0 together form another odd cycle in G; the rest of the argument isstraightforward. ▯
1074 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
3. Graphs with one odd cycle length. This section is devoted to the studyof graphs with jLj ¼ 1; we prove auxiliary Theorem 3.1 here, which strengthensTheorem 1.1 for this case.
THEOREM 3.1. LetG be a graph with jLðGÞj ¼ 1 not containing a K 4; let C be an oddcycle in G. Then any proper 3-coloring of C can be extended to a proper 3-coloring of G.
Proof. We may clearly assume that G is 2-connected. Most of the proof consists ofcollecting further structural information about G; we include these observations asClaims 1–4. After describing the structure of G sufficiently, we will be able to findthe desired 3-coloring directly.
At the beginning, we use Lemma 2.5(1) to constrain the following configuration,shown in Figure 3.1.
CLAIM 1. Let Pi ≔ xi · · · y, i ∈ Z3, be nontrivial paths inG such that V ðPi ∩ CÞ ¼fxig and V ðPi ∩ PjÞ ¼ fyg for any j ∈ Z3, j ≠ i. Let Pi
0 denote the subpath of C be-tween xi−1 and xiþ1 not containing xi. Then kPik ¼ kPi
0k.Proof. For brevity, let αi ≔ kPik and βi ≔ kPi
0k. We consider each of the three C -paths PiPiþ1. By Lemma 2.5(1), its length αi þ αiþ1 must be equal to either βi þ βiþ1 orβiþ2. Next, αi ≠ 0 by the nontriviality of Pi. Finally, β0 þ β1 þ β2 ¼ jC j; therefore, thissum is odd. Hence, αi and βi satisfy the assumptions of Lemma 2.3, which yields αi ¼ βi
for all i. ▯By Lemma 2.4(1) we already know that odd cycles in G cannot be disjoint from C .
Further restrictions for the position of odd and also even cycles with respect to C areexpressed as the two subsequent claims.
CLAIM 2. Every cycle D in G with exactly two vertices x0 and x1 in common with C ,such that D ⊆ M for some C-bridge M , is even. Moreover, kD0k ¼ kD1k, where D0 andD1 are the two subpaths of D joining x0 and x1.
Proof. Let c0 and c1 denote the lengths of the two subpaths of C delimited by x0and x1. As D is a subgraph of the C -bridgeM , by Lemma 1.3(1, 2) each Di, i ¼ 0, 1, hasan inner vertex, andM contains a path connecting these two vertices and avoiding bothx0 and x1. One can take its subpath P ≔ y0 · · · y1 such that V ðP ∩ DiÞ ¼ fyig. Notethat P is nontrivial. For brevity, let the lengths of the paths P, x0Diyi, and x1Diyi bedenoted by d, ai, and bi, respectively. See Figure 3.2.
We proceed by contradiction; assume jjD0jj ≠ jjD1jj. Then by Lemma 2.5(1) weobtain, without loss of generality, that
We consider two other C -paths, x0D0y0Py1D1x1 and x0D1y1Py0D0x1. These must sa-tisfy Lemma 2.5(1), too; i.e., the following holds:
a0 þ b1 þ d ¼ cj0 and a1 þ b0 þ d ¼ cj1
for some j0, j1 ∈ f0; 1g. Summing the formulas and using the equalities (3.1), we con-clude that
c0 þ c1 þ 2d ¼ cj0 þ cj1 :
If j0 ≠ j1, then d ¼ 0; that is impossible since P is nontrivial. Hence, j0 ¼ j1, butthen the equation turns into c0 þ c1 ¼ 2ðcj0 − dÞ, which is a parity contradiction asC is odd. ▯
CLAIM 3. Let D be a cycle in G; let P0, P1, and P2 be pairwise disjoint C–D pathssuch that V ðC ∩ DÞ ⊆ V ðP0 ∪ P1 ∪ P2Þ. Then neither of the following holds:
(1) At most one of the paths P0, P1, and P2 is trivial.(2) Exactly two of the paths P0, P1, and P2 are trivial, and D ⊆ M for some C-
bridge M .Proof. The appropriate end-vertices of P0, P1, and P2 split each of the cycles C and
D into three subpaths; let these be denoted by Ci, i ¼ 0, 1, 2, and Di, respectively, insuch a way that V ðPi ∩ CiÞ ¼ ∅ and V ðPi ∩ DiÞ ¼ ∅. Further, let ci ≔ kCik. SeeFigure 3.3(a).
First assume part (1) to be true. Then let’s say P0 is the only trivial path if there isone at all. We take the paths P0D1, P1D0, and P2 and apply Claim 1, obtaining thatkP2k ¼ c2. Doing the same for the paths P0D2, P2D0, and P1 yields kP2k þ kD0k ¼ c2.It follows that kD0k ¼ 0, a contradiction to the assumptions.
Now suppose that assertion (2) holds. Without loss of generality, P0 is the onlynontrivial path; the situation is shown in Figure 3.3(b). Then by applying Claim 1to the paths P0,D1, andD2, we obtain that kP0k ¼ c0, kD1k ¼ c2, and kD2k ¼ c1. Next,by Claim 2, we have kD0k ¼ kD1k þ kD2k. However, this means that there is a cycleP0D2D0C1 of length c0 þ 3c1 þ c2 in G, which is an odd cycle longer than C . ▯
We are now able to describe the structure of the C -bridges. We remark that everyattachment of an arbitrary C -bridge M belongs to a unique block of M , since byLemma 1.3(2), it is not a cut-vertex of M .
FIG. 3.2. The proof of Claim 2. The indicated lengths of paths are enclosed in brackets.
1076 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
CLAIM 4. Let M be a C-bridge. Then(1) M has at most three attachments;(2) if M has precisely three attachments, then each of these vertices lies in a dif-
ferent block of M ;(3) M is a bipartite graph.Proof. We start by proving part (1). Assume to the contrary that M is a C -bridge
with more than three attachments; take four of them, xi, i ∈ Z4, located on C in a cyclicorder. These vertices split C into four paths denoted by Ci in such a manner thatCi ¼ xi · · · xiþ1. Let ci ≔ kCik.
By Lemma 1.3(1, 2), there exists a C -path P inM connecting x0 with x1 and havingan inner vertex. Further, Lemma 1.3(2) asserts that this vertex is joined to x2 by apath not containing any vertex of C \ x2. Taking its part P2 ≔ x2 · · · y such thatV ðP2 ∩ PÞ ¼ fyg and putting Pi ≔ xiPy, i ¼ 0, 1, we get the situation of Claim 1.However, there is one more attachment x3. Again, x3 must be connected with y bya path avoiding all vertices ofC \ x3; let P3 ≔ x3 · · · z be its subpath such thatV ðP3Þ ∩V ðP ∪ P2Þ ¼ fzg. There are four possibilities regarding the position of z in P ∪ P2:either z ¼ y or z ∈ V ðPi \ yÞ for some i ∈ f0; 1; 2g.
Let z ¼ y or z ∈ V ðP2 \ yÞ; the latter situation is shown in Figure 3.4(a). ApplyingClaim 1 subsequently to the triples of paths fP0; P1; P2g and fP0; P1; P3zP2yg, we ob-tain that kP0k ¼ c1 and kP0k ¼ c1 þ c2, respectively. This implies that c2 ¼ 0, a con-tradiction. By symmetry, we can deal with the case z ∈ V ðP0 \ yÞ similarly. Thus, z lieson P1 \ y. First let ai, i ¼ 0; : : : ; 3, denote the lengths of the paths P0, P 0
1 ≔ x1P1z, P2,and P3, respectively. See Figure 3.4(b) for the notation. Now, using Claim 1 twice for thetriples of paths fP0P1z; P
01; P3g and fP2P1z; P
01; P3g, we obtain, in particular, that
a1 ¼ c3, a3 ¼ c0, a1 ¼ c2, and a3 ¼ c1; therefore, c0 ¼ c1 and c2 ¼ c3. By symmetry,one can show that also c2 ¼ c1 and c0 ¼ c3. It follows that c0 ¼ c1 ¼ c2 ¼ c3, and there-by, jC j ¼ 4c0. However, this is impossible as C is odd.
Let us focus on statement (2); we refer to the three attachments ofM as u, v, andw.First observe the following:
There is no cycle in M passing through two of the attachments and
If there were such a cycle D with, say, w ∈= V ðDÞ, then by Lemma 1.3(2) we could find aw–D path in M \ fu; vg, and thus obtain the configuration excluded by Claim 3(2).
We proceed by contradiction now: let u and v belong to a single block B of M . ByLemma 1.3(1), u and v are nonadjacent inM . Thus, B is not a K2; it is 2-connected, andthere is a cycle D containing both u and v. Observation (3.2) implies that w ∈ V ðDÞ aswell. Consider the graphM \ w and the pathD \ w. By Lemma 1.3(1, 2), u is not an end-vertex of D \ w nor a cut-vertex of M \ w; the latter means that u belongs to a uniqueblock Bu of M \ w. See Figure 3.5.
It is easy to see that, in general, every block of an arbitrary graph H and any pathPH in H either are disjoint or intersect in a path (possibly trivial) whose end-vertices areeither cut-vertices ofH or end-vertices of PH . Applying this to the graphM \ w, its blockBu, and the pathD \ w, we deduce that P ≔ ðD \ wÞ ∩ Bu is a nontrivial path with end-vertices u1 and u2 distinct from u. It follows that Bu is not aK 2; it is 2-connected, and byTheorem 1.5, there is a path P 0 ≔ u1 · · · u2 in Bu avoiding u.
FIG. 3.4. The proof of Claim 4(1). The paths P0, P1, and P2 are printed in bold.
FIG. 3.5. The proof of Claim 4(2). The bold lines and edges represent the cycle contradictingobservation (3.2).
1078 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
Note now that v ∈= V ðBuÞ. Otherwise, since Bu is 2-connected, we would easily con-tradict observation (3.2). Hence, by replacing P with P 0 in the cycleD, we obtain a cyclepassing through v and w but not u—a contradiction to observation (3.2) again.
We prove assertion (3). So far we know by parts (1) and (2) that every block of Mcontains at most two attachments of M . Consequently, every cycle D in M has at mosttwo vertices in common with C . If jD ∩ C j ¼ 2, Claim 2 implies that D is even; ifjD ∩ C j ≤ 1, it must be even as well by Lemma 2.4(1). Therefore, M contains noodd cycle; it is bipartite. ▯
We are ready to finish the proof of Theorem 3.1. Let cC denote a prescribed proper3-coloring of C . Clearly, it suffices to show that we can extend cC to the graph C ∪ Mfor every C -bridge M . By Claim 4(3, 1) and the 2-connectedness of G, any such M isbipartite, it has two or three attachments, and these are pairwise nonadjacent in M byLemma 2.5(1). Thus, when M has exactly two attachments, using Lemma 2.1 we aredone. Assume therefore thatM has precisely three attachments, xi, i ¼ 0, 1, 2. We recallthat each xi lies in a unique block Bi of M ; by Claim 4(2), these blocks are pairwisedifferent.
Consider the block graph BðM Þ of M . Observe that every leaf of BðMÞ contains xifor some i; otherwise,G would have a cut-vertex. Hence,BðMÞ is a tree with two or threeleaves. Suppose first that the former case holds; the subsequent notation is shown inFigure 3.6(a). Then BðMÞ is a path with, say, B0 and B1 as its end-vertices. The blockB2 contains exactly two cut-vertices ofM , denoted by aj, j ¼ 0, 1, in such a way that ajlies on the path Bj · · · B2 in BðM Þ. Note that both a0 and a1 are distinct from x2, andconsequently, B2 ≠ K2. Thus, we can apply Lemma 2.2 to the vertices a0, a1, and x2, inthis order, and the block B2, obtaining a cycle D ⊆ B2 passing through both a0 and a1along with an x2–D path P2 in B2 (possibly trivial). Further, there clearly exists a non-trivial C–B2 path Pj, j ¼ 0, 1, connecting xj with aj in M . But then all three paths Pi,i ¼ 0, 1, 2, together with the cycles C and D form the configuration excluded byClaim 3(1)
Therefore, BðM Þ has precisely three leaves; it consists of three nontrivial paths shar-ing a single end-vertex x. Assume that x is a block ofM . We may then proceed similarlyas in the preceding case, with x in place of B2. Obviously, the block x is 2-connected andcontains exactly three cut-vertices denoted by ai in such a manner that ai lies on the
FIG. 3.6. The proof of Theorem 3.1. The configurations given by Lemma 2.2 are depicted in bold.
path xi · · · x in BðM Þ. See Figure 3.6. Applying Lemma 2.2 to a0, a1, and a2 and theblock x, we get a cycle D containing a0 and a1 together with an a2–D path P 0
2; both aresubgraphs of x. Next, we can find three (pairwise disjoint and nontrivial) C–x paths Pi,each connecting xi with ai in M . Then, however, Claim 3(1) used for the paths P0, P1,and P2P2
0 and the cycles C and D yields a contradiction again.Hence, x is a cut-vertex ofM . LetMi denote the subgraph ofM corresponding to the
path xi · · · x in BðMÞ. If xix ∈ EðGÞ for all i, then Claim 1 would force that C ¼ K 3;i.e., there would be a K4 in G, which is excluded by assumption. Therefore, without lossof generality, x0x is not an edge of G, and hence, we can extend properly the 3-coloringcC to the graph C ∪ M first by coloring x with a color of cC distinct from both cC ðx1Þand cC ðx2Þ and then by using Lemma 2.1 for each Mi. ▯
4. Graphs with two odd cycle lengths. The aim of this section is to prove themain result, Theorem 1.2. As a preliminary step, we focus on the class of 4-critical graphswith L ¼ fk; kþ 2g, k ≥ 5, obtaining useful structural constraints in the form ofProposition 4.1 and Corollary 4.2.
Clearly, Lemma 2.4(1) applies to the graphs in question. With the stronger assump-tion of 4-criticality and the use of Theorem 3.1, we are able to strengthen it to thefollowing statement.
PROPOSITION 4.1. LetG be a 4-critical graph with LðGÞ ¼ fk; kþ 2g, k ≥ 5. Then notwo odd cycles in G are disjoint.
Proof. Assume to the contrary that there are two such cyclesC1 andC 2 inG. Then,by Lemma 2.4(2a), jC1j ¼ jC2j ¼ k. We show first that there is a vertex x of C 1 whichlies on no C 1–C 2 path. For this, we discuss two cases.
If G contains three pairwise disjoint C 1–C 2 paths Pi, i ¼ 1, 2, 3, we choose x as oneof the vertices in V ðC 1Þ \ fx1; x2; x3g, where xi ≔ V ðPi ∩ C 1Þ. See Figure 4.1(a). Let usprove that x indeed has the desired property. Assume to the contrary that there is aC 1–C 2 path P in G containing the vertex x. By Lemma 2.4(2d), P is not disjoint fromall Pi. Note that each Pi is a single edge, as forced by Lemma 2.4(2b). Consequently,without loss of generality, P is disjoint from P2 and P3, and its other end-vertex lying onC 2 coincides with that of P1. Now we apply Lemma 2.4(2c) subsequently to the triples of
FIG. 4.1. The proof of Proposition 4.1: Finding the vertex x.
1080 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
paths fP1; P2; P3g and fP;P2; P3g; it implies that dC1ðx2; x1Þ ¼ dC1
ðx2; xÞ anddC1
ðx3; x1Þ ¼ dC1ðx3; xÞ. As C 1 is odd, it follows that x ¼ x1, which is a contradiction
to the choice of x.On the other hand, let there be no three pairwise disjoint C 1–C 2 paths inG. AsG is
4-critical, it is 2-connected; hence, by Theorem 1.5, there are two disjoint C 1–C 2 pathsP1 and P2. Let xi, i ¼ 1, 2, denote the end-vertex of Pi lying on C 1, andlet a be the distance of the other end-vertices of Pi in C 2. See Figure 4.1(b). ByLemma 2.4(2b), both Pi have length 1, and dC1
ðx1; x2Þ ¼ a. Consider now anyC 1–C 2 path P3 with an end-vertex y on C1 distinct from both xi. By assumption,P3 intersects, say, P1 in a vertex on C 2, and it is disjoint from P2. Then by Lemma 2.4(2b) again, applied to P2 and P3, we obtain that also dC1
ðx2; yÞ ¼ a. It follows that thereare at most four end-vertices of C1–C2 paths on C 1, as shown in Figure 4.1(b). ButjC1j ≥ 5; we choose x as any of the remaining vertices.
So, in either case, the vertex x exists. As G is 4-critical, the degree of x is at least 3.Hence, G has an edge incident with x not belonging to C 1 and thus contained in some(C 1 ∪ C 2)-bridge H . By Lemma 1.3(2), any two attachments of H are joined by a pathin H avoiding the other attachments. Therefore, since x lies on no C 1–C2 path, all theattachments of H must be on C 1.
We show that the existence of such a bridge leads to a contradiction. Take thegraphs G1 and G2 such that V ðG1Þ ¼ ðV ðGÞ \ V ðHÞÞ ∪ V ðC 1Þ, EðG1Þ ¼ EðGÞ \EðHÞ, andG2 ¼ H ∪ C 1. See Figure 4.2. The graphG1 is a proper subgraph ofG; there-fore, by the 4-criticality of G, it has a proper 3-coloring, c1. If G2 contained a cycle oflength kþ 2, then this cycle together with C 2 would be a pair of disjoint odd cycles inG,contradicting Lemma 2.4(1). Thus, jLðG2Þj ¼ 1, and by Theorem 3.1, we can extend thecoloring c1 of C1 to a proper 3-coloring c2 of G2. Clearly, c1 ∪ c2 is then a proper3-coloring of G. ▯
By the combination of Proposition 4.1 and Lemma 1.4, we easily infer the nextassertion.
COROLLARY 4.2. Every 4-critical graph with L ¼ fk; kþ 2g, k ≥ 5, is 3-connected.Proof. Suppose that, on the contrary, there exists a graph G satisfying the assump-
tions but containing a 2-cut fu; vg, which splits G into two graphs G1 and G2.Then Lemma 1.4 applied first to u and any of the vertices of G1 and next to v andany of the vertices of G2 gives two odd cycles C 1 and C 2 such that V ðC 1Þ ⊆ V ðG1Þ ∪fvg and V ðC 2Þ ⊆ V ðG2Þ ∪ fug. These cycles are disjoint, which contradictsProposition 4.1. ▯
Now we are ready to prove the main result.
FIG. 4.2. The proof of Proposition 4.1: Considering the bridge H . The light and dark gray regions re-present G1 and H , respectively; the bold dotted line bounds G2.
Proof of Theorem 1.2. We proceed by contradiction. Assuming that the theorem isfalse, we take a minimal counterexample with respect to inclusion, which clearly is a4-critical graph G with LðGÞ ¼ f5; 7g. Similarly to the proof of Theorem 3.1, we firstinvestigate the structure of G by establishing Claims 1–6. Using these results, we finishby showing that G is, in fact, 3-colorable.
To start, fix C ≔ v0v1 · · · v6 to be one of the 7-cycles in G. For convenience, wedefine the following notions. Let x be a vertex of G \ C . Then each of its neighbors lyingon C is a friend of x; the set of all friends of x is denoted by FðxÞ. We make an easyobservation now.
CLAIM 1. The following holds:(1) For any vertex x of G \ C , FðxÞ is an independent set in G.(2) Let x and y be two adjacent vertices of G \ C . Then FðxÞ ∩ FðyÞ ¼ ∅, and
FðxÞ ∪ FðyÞ is an independent set in C .Proof. If x and y in part (2) have friends f 1 and f 2, respectively, that are
adjacent in C , then there is a C -path f 1xyf 2 of length 3 with its end-vertices at distance1 in C , which is a contradiction to Lemma 2.5(2). The rest is clear since G has notriangles. ▯
By Corollary 4.2, G is 3-connected. Using this and the fact that jC j ¼ 7, we are ableto prove the following statement, crucial for the rest of the argument.
CLAIM 2. Every vertex of G \ C has at least one friend.Proof. Suppose the contrary; i.e., there exists a vertex x of G at distance at least 2
from C . As G is 3-connected, by Theorem 1.5 it contains three internally disjoint x–Cpaths Pi, i ∈ Z3, with distinct end-vertices xi on C . These paths constitute three C -paths P 0
i ≔ Piþ1Piþ2, all of which have length at most 6 by Lemma 2.5(2a). By assump-tion, jjPijj ≥ 2; thereby jjPijj ≤ 4.
We discuss all the possible forms of the multiset P ≔ fjjPijj; i ∈ Z3g. First, withoutloss of generality, let P0 be of length 4. It follows that jjP1jj ¼ jjP2jj ¼ 2, and P 0
i havelengths 4, 6, and 6, respectively. But then Lemma 2.5(2) applied to each P 0
i forces thatdC ðx0; x1Þ ¼ dC ðx0; x2Þ ¼ 1 and dC ðx1; x2Þ ¼ 1 or 3, which is clearly not possible. Bysimilar reasoning, we exclude the cases P ¼ f3; 3; 3g and P ¼ f3; 3; 2g; hence, P mustequal either f3; 2; 2g or f2; 2; 2g. Thus, we have proven the following assertion so far:
For any vertex y of G at distance at least 2 from C; there exist two
internally disjoint y–C paths of length 2 in G.ð4:1Þ
We continue by showing that the two remaining cases for P cannot occur either. Most ofthe time, the subsequent proof consists of applying Lemma 2.5(2a) or Lemma 2.5(2) tovarious C -paths; we omit explicit reference to these particular assertions for brevity.
We focus on the case P ¼ f3; 2; 2g. Without loss of generality, let jjP0jj ¼ 3,jjP1jj ¼ jjP2jj ¼ 2, and x0 ¼ v0. Then, considering the C -paths P 0
1 and P 02, both of
length 5, we see that dC ðx0; x1Þ ¼ dC ðx0; x2Þ ¼ 2. Hence, say that x1 ¼ v2 andx2 ¼ v5. Let us label the inner vertices of P0, P1, and P2 in such a way thatP0 ¼ v0uy0x, P1 ¼ v2y1x, and P2 ¼ v5y2x. See Figure 4.3(a).
By the 4-criticality of G, the degree of u is greater than 2, and therefore, there is atleast one neighbor v of u other than v0 and y0. This vertex is distinct from both x and v1;otherwise, a triangle would occur inG. Next, it does not equal v2, or else we would have aC -path P2P0uv2 of length 5 with its end-vertices at distance 3 in C . If v ¼ v3, then Gwould contain a C -path P1P0uv3 of length 5 with end-vertices adjacent in C . Finally, vcannot be y1; otherwise, the C -path P2P0uy1v2 of length 6 with its end-vertices
1082 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
nonadjacent in C would exist in G. As the vertices v4, v5, v6, and y2 can be treated in acorresponding manner by symmetry, we conclude that v ∈= V ðC ∪ P0 ∪ P1 ∪ P2Þ.
We prove that v has no neighbor in C ∪ P0 ∪ P1 ∪ P2 \ fu; xg. First, we show thatv has no friends. Assume the contrary; without loss of generality, we can choose a friendw of v different from v5. Then there is a C -path P2P0uvw of length 6 in G. Conse-quently, w ¼ v4 or w ¼ v6. In both cases, however, we obtain a contradiction, consider-ing another C -path P1P0uvw of length 6. Next, vy1 ∈= EðGÞ since otherwise we wouldget a C -path P2P0uvy1v2 of length 7. The vertex y2 is not adjacent to v either, by sym-metry. Finally, vy0 ∈ EðGÞ would create a triangle in G.
As v has no friends, by statement (4.1) there exist two internally disjoint v–C pathsof length 2. At least one of them, P, does not pass through u. It also avoids x because thedistance between x and C is greater than 1 by assumption. Hence, P is disjoint fromP0 ∪ P1 ∪ P2 \ C by the previous discussion. Without loss of generality, the other end-vertex of P is distinct from v5. But then G contains a C -path P2P0uvP of length 7, acontradiction.
It remains to discuss the case P ¼ f2; 2; 2g; we proceed similarly as above. Thistime, the C -path Pi
0 of length 4 forces that dC ðxiþ1; xiþ2Þ ¼ 1 or 3 for every i. Onecan easily check that this implies, up to symmetry, x0 ¼ v0, x1 ¼ v3, and x2 ¼ v4.Let u, y1, and y2 denote the inner vertex of P0, P1, and P2, respectively. SeeFigure 4.3(b).
Since the degree of u is at least 3, u has a neighbor v distinct from both v0 and x. Thevertex v cannot equal v1 or y1 because there are no triangles in G. If uv2 ∈ EðGÞ, thenP2xuv2 would be a C -path of length 4 with its end-vertices at distance 2 in C . Also,v ≠ v3; in such a case G would contain a 9-cycle v4P2P1v3uv0v6v5v4. By symmetry,we can exclude the vertices y2, v4, v5, and v6 as well. Hence, v ∈= V ðC ∪ P0 ∪ P1 ∪ P2Þ.
We claim that v has no neighbors in C ∪ P0 ∪ P1 ∪ P2 \ u. First, letw be a friend ofv; without loss of generality, w ≠ v4. Then there is a C -path P2xuvw of length 5, andconsequently, dC ðv4; wÞ ¼ 2. It follows that dC ðv3; wÞ ¼ 1 or 3, but this is impossibledue to the C -path P1xuvw also of length 5. Hence, v is without friends. Next, ifvy1 ∈ EðGÞ, we would have a C -path P2xy1vuv0 of length 6 with end-vertices nonad-jacent in C . The vertex y2 can be excluded similarly by symmetry. Finally, vx ∈ EðGÞwould imply a triangle in G.
As we have just shown, the distance between v andC is 2; again, statement (4.1) canbe applied to v. Using it and the preceding analysis, we see that there exists a v–C pathP ≔ v · · · w of length 2 which is disjoint from P1 ∪ P2 ∪ P3 \ C . We may assume thatw ≠ v4. The C -path P2xuvP of length 6 then forces that w ¼ v3 or w ¼ v5. The lattercase cannot occur, as seen by considering anotherC -path P1xuvP of length 6. Therefore,w equals v3. Then, however, v4P2P1Pvuv0v6v5v4 is an 11-cycle, which is a contradictionto the assumption about LðGÞ. ▯
Having established the preceding claim, we can restrict significantly the structureof G \ C .
CLAIM 3. G \ C is a forest.Proof. Assume that the claim is false; let D be a cycle in G \ C . We take two arbi-
trary verticesw0 andw1 consecutive onD. By Claim 2, both the vertices have friends; letg0 and g1 be a friend of w0 and w1, respectively. The vertices g0 and g1 are distinct byClaim 1(2), and therefore, the edges g0w0 and g1w1 together with the nontrivial part ofD between w0 and w1 constitute a C -path Q of length jDj þ 1. Lemma 2.5(2a) thenimplies that jDj ≤ 5. Thus, by Lemma 2.4(1) and the assumption about LðGÞ,jDj ¼ 4. Hence, Q is of length 5, and Lemma 2.5(2) forces that
dC ðg0; g1Þ ¼ 2:ð4:2ÞNow let ui, i ∈ Z4, denote the vertices of D in a cyclic order. Suppose first that there
are two vertices opposite on D, say, u0 and u2, with distinct friends f 0 and f 2, respec-tively; we pick a friend f j of uj, j ¼ 1, 3. By (4.2), we see that dC ðf i; f iþ1Þ ¼ 2 for all i.Consequently, dC ðf 0; f 2Þ ¼ 3, and the vertices f 1 and f 3 coincide. The situation is shownin Figure 4.4(a). But then the cycle consisting of the path f 0u0u1f 1u3u2f 2 and the ap-propriate part of C has length 9, a contradiction.
Thus, we must have the following arrangement: every ui has exactly one friend f i, itis f j ¼ f jþ2, j ¼ 0; 1, and dC ðf 0; f 1Þ ¼ 2. See Figure 4.4(b). Now, asG is 3-connected, byTheorem 1.5 there exist three pairwise disjointD–C paths inG. At least one of them, P,has its end-vertex v on C distinct from both f 0 and f 1. We may assume that the otherend-vertex of P is u0 by symmetry. The path P is not a single edge; hence, consideringthe C -path P 0 ≔ Pu0u1u2u3f 1 and Lemma 2.5(2a), we infer that jjPjj ¼ 2. Then, how-ever, Lemma 2.5(2) applied subsequently to the C -paths P 0 of length 6 and Pu0u1u2f 0of length 5 asserts that dC ðv; f 1Þ ¼ 1 and dC ðv; f 0Þ ¼ 2, respectively, which isimpossible. ▯
FIG. 4.4. The proof of Claim 3.
1084 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
We proceed to a detailed analysis of the structure and mutual position of the C -bridges. Let us start with an auxiliary observation.
CLAIM 4. Let wi, i ¼ 1; : : : ; 4, be four consecutive vertices of C ; let x and y be twodistinct vertices of G \ C . Suppose that fw1; w3g ⊆ FðxÞ. Then fw2; w4g ⊈ FðyÞ.
Proof. If the contrary held, w1xw3w2yw4Cw1 would be a 9-cycle in G. ▯Claim 1(1) and the fact that jC j ¼ 7 imply that any vertex x of G \ C has at most
three friends. Moreover, if it has three friends f 0, f 1, and f 2, then the multisetfdC ðf i; f iþ1Þ; i ∈ Z3g equals f2; 2; 3g. Extending this, the subsequent claim describescompletely the C -bridges involving three-friend vertices.
CLAIM 5. The following holds:(1) Every C-bridge containing a vertex with three friends is a K 1;3.(2) There exists an i ∈ Z7 such that every C-bridge of the form K 1;3 intersects C in
either fvi; viþ2; viþ5g or fvi; viþ3; viþ5g.Proof. We begin with part (1). To the contrary, let M be a C -bridge not equal
to a K 1;3 that contains a three-friend vertex x. Then jM \ C j > 1, and hence, byLemma 1.3(2), there is a vertex in M \ C adjacent to x. This vertex has a friend f ,as forced by Claim 2; f and the three friends of x constitute, by Claim 1(2), a 4-elementindependent set in C , which is a contradiction.
We proceed to show part (2). By the discussion immediately preceding Claim 5,every C -bridge of the form K1;3 intersects C in fvi; viþ2; viþ5g for some i ∈ Z7. Thus,when all such C -bridges have the same attachments, we are done.
On the other hand, letM 1 andM 2 be two C -bridges, each equal to a K1;3, such thatV ðM 1 ∩ CÞ ≠ V ðM 2 ∩ CÞ. Without loss of generality, the attachments ofM 1 are v0, v2,and v5. By assumption, the attachments of M 2 must be vi, viþ2, and viþ5 for somei ∈ Z7 \ f0g. But if i ∈ f1; 3; 4; 6g, we would get a configuration excluded by Claim 4.The remaining two cases, i ¼ 2 and i ¼ 5, are identical up to symmetry; we may there-fore suppose that i ¼ 5. The situation is shown in Figure 4.5.
Consider now aC -bridgeM of the formK1;3 such thatV ðM ∩ CÞ ≠ V ðMj ∩ CÞ forboth j ¼ 1, 2. The preceding analysis repeated forM andMj with j fixed implies thatMhas exactly two attachments in common with Mj, and these are at distance 2 in C .However, that obviously cannot occur for both j at the same time, as can be seen inFigure 4.5; this is a contradiction. Thus, every C -bridge in question has the same at-tachments as either of Mj, and the claim is proven. ▯
Now we characterize the other C -bridges.CLAIM 6. Every C-bridge containing no vertex with three friends is a K2.
FIG. 4.5. The proof of Claim 5(2). The C-bridge M 1 is distinguished with boldface.
Proof. Assume to the contrary that G has a C -bridgeM without three-friend verti-ces such that the graphM 0 ≔ M \ C is nonempty. By Claim 3 and the connectedness ofa C -bridge, M 0 is a tree. Next, every vertex of M 0 has degree at least 3 in M by the4-criticality of G; since it has at most two friends by assumption, its degree in M 0 isnonzero. Therefore, M 0 is not trivial and, thus, has at least two leaves. Note that everyleaf of M 0 has precisely two friends.
We take an arbitrary pair of distinct leaves x and y of M 0; let FðxÞ ¼ fx1; x2g andFðyÞ ¼ fy1; y2g. The leaves are joined by a unique path P in M 0; we refer to its lengthas l. Further, let dx ≔ dC ðx1; x2Þ and dy ≔ dC ðy1; y2Þ, and let D be the setfdC ðxi; yjÞ; i; j ¼ 1; 2g. Without loss of generality, we may assume that x1 ¼ v0 andx2 ¼ vdx . See Figure 4.6(a) for the situation and notation.
Considering Claim 1(1) and the particular length of C , we see that dx, dy ∈ f2; 3g.Next, observe the following basic properties of the set D:
0 ∈ D ⇒ fdx; dyg ⊆ D;ð4:3ÞjDj ≥ 2:ð4:4Þ
Statement (4.3) is obvious. To prove assertion (4.4), suppose the contrary, D ¼ fag.Clearly, a ≠ 0 by statement (4.3), and hence, the friends of x and y are pairwise distinct.But then both y1 and y2 must be at distance a from both x1 and x2 in C , which is im-possible as there is at most one vertex with this property.
Now, every two distinct friends xi and yj are joined by a C -path of length lþ 2consisting of the edges xix and yjy and the path P. Thus, first, Lemma 2.5(2) liststhe possible nonzero elements of D when l is known; for brevity, we will use this factwithout further mention. Second, since by statement (4.3), such a pair of distinct friendsindeed exists, it must be l ≤ 4 by Lemma 2.5(2a). On the other hand, Claim 1(2) impliesthat l ≥ 2. Let us consider each of the three possible values for l separately.
First, assume that l ¼ 4. Then D ⊆ f0; 1g. Hence, dx ∈= D, and thus, by assertion(4.3), D ¼ f1g. This contradicts statement (4.4), however.
If l ¼ 3, then D ⊆ f0; 2g; therefore, D ¼ f0; 2g by statement (4.4). Assertion(4.3) then forces that dx ¼ dy ¼ 2. As 0 ∈ D, it is, say, x1 ¼ y1. If x2 and y2 weredistinct, the vertex y2 would coincide with v5, but then dC ðx2; y2Þ ¼ 3, which is a
FIG. 4.6. The proof of Claim 6.
1086 TOMÁŠ KAISER, ONDŘEJ RUCKÝ, AND RISTE ŠKREKOVSKI
contradiction. Thus, x2 ¼ y2. Now we consider the inner vertex z of P adjacent to x.By Claim 2, this vertex has a friend z 0; Claim 1(2) applied to x and z implies that,without loss of generality, z 0 ¼ v5. Consequently, G contains a C -path x1yPzz
0
of length 4 with end-vertices at distance 2 in C , which, however, contradictsLemma 2.5(2).
Therefore, l equals 2, and hence,D ⊆ f0; 1; 3g. Focus on the unique inner vertex z ofP; it has a friend z 0 by Claim 2. As z is adjacent to both x and y, we can use Claim 1(2)twice to see that z 0 is the unique friend of z, it is distinct from the friends of x and y, andthe sets F1 ≔ fx1; x2; z 0g and F2 ≔ fy1; y2; z 0g are both independent in C . Now supposethat x or y has its friends at distance 2 in C ; say, it is dx ¼ 2. Then, by statement (4.3),D ⊆ f1; 3g, which means that fy1; y2g ⊆ fv1; v3; v6g. Claim 4 subsequently forces thatfy1; y2g ¼ fv3; v6g, but then there is no vertex onC satisfying the conditions imposed onz 0—a contradiction. Thus, dx ¼ dy ¼ 3. By the independence of F1 and F2 in C again, itfollows that fx1; x2g ¼ fy1; y2g ¼ fv0; v3g and z 0 ¼ v5.
We sum up the facts inferred so far. Every two leaves of M 0 are at distance 2 in M 0;i.e., M 0 is a star. Next, any two, hence all, leaves ofM 0 have the same (two) friends, andthese are at distance 3 in C ; without loss of generality, they coincide with v0 and v3.Finally, the central vertex z of M 0 has a unique friend v5. See Figure 4.6(b). But thestructure of M then contradicts the 4-criticality of G because any proper 3-coloringc of G \ M 0 can be extended to G as follows. We color all the leaves of M 0 with a colorc1 of c distinct from both cðv0Þ and cðv3Þ, and we assign a color c2 of c equal to neither c1nor cðv5Þ to z. ▯
We can finally derive a contradiction by proving that G is 3-colorable. Without lossof generality, by Claims 5 and 6 we may partition all the C -bridges into three sets S1, S2,and K such that S1 and S2 contain only K1;3s intersecting C in fv1; v3; v6g andfv1; v4; v6g, respectively, and K consists of single edges. Next, we take a proper coloringc of C with colors 1, 2, and 3 such that the color classes are fv0; v5g, fv2; v4g, andfv1; v3; v6g, in this order. See Figure 4.7. The coloring c has the property that thereis only one pair of equally colored vertices at distance 3 in C—the vertices v3 andv6; using Lemma 2.5(2b) one observes that
if v3v6 ∈= K; then c is a proper coloring of C ∪[
K.ð4:5ÞNow we discuss several cases. If S1 ≠ ∅, then the assumption of statement (4.5) is
satisfied; otherwise, there would be a triangle in G. As c can be easily extended to anyelement of S1 ∪ S2 by coloring its central vertex with 1, we conclude that G is indeed
FIG. 4.7. The proof of Theorem 1.2: The 3-colorability of G when S1 ≠ ∅. The colors of vertices are writ-ten in brackets; the dotted edges and white vertices represent the possibly nonexistent parts of G.
3-colorable. The case S2 ≠ ∅ is handled analogously by symmetry. Finally, there re-mains the possibility that both S1 and S2 are empty sets. If K is empty as well, thereis nothing to prove. Otherwise, without loss of generality, v2v6 ∈ K . Then v3v6 ∈= Kclearly, and we are done by observation (4.5) again.
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