Abstract If we want to apply Galvin’s kernel method to show that a graph G satisfiesa certain coloring property, we have to find an appropriate orientation of G. Thismotivated us to investigate the complexity of the following orientation problem. Theinput is a graph G and two vertex functions f, g : V (G) → N. Then the question iswhether there exists an orientation D of G such that each vertex v ∈ V (G) satisfies∑
u∈N+D (v) g(u) ≤ f (v). On one hand, this problem can be solved in polynomial time
if g(v) = 1 for every vertex v ∈ V (G). On the other hand, as proved in this paper,the problem is NP-complete even if we restrict it to graphs which are bipartite, planarand of maximum degree at most 3 and to functions f, g where the permitted valuesare 1 and 2, only. We also show that the analogous problem, where we replace gby an edge function h : E(G) → N and where we ask for an orientation D suchthat each vertex v ∈ V (G) satisfies
∑e∈E+
D(v) h(e) ≤ f (v), is NP-complete, too.Furthermore, we prove some new results related to the ( f, g)-choosability problem,or in our terminology, to the list-coloring problem of weighted graphs. In particular,
Research supported in part by the DAAD and by the Hungarian Scientific Research Fund, grant OTKAT-81493.
M. Stiebitz (B)Technical University of Ilmenau, PF 10 05 65, 98684 Ilmenau, Germanye-mail: [email protected]
Z. TuzaAlfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences,Reáltanado u. 13-15, Budapest 1085 Hungary
Z. TuzaDepartment of Computer Science and System Technology,University of Pannonia, Veszprem, Hungary
M. VoigtUniversity of Applied Sciences Dresden, Friedrich-List-Platz 1, 01069 Dresden, Germany
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we use Galvin’s theorem to prove a generalization of Brooks’s theorem for weightedgraphs. We show that if a connected graph G has a block which is neither a completegraph nor an odd cycle, then G has a kernel perfect super-orientation D such thatd+
There are several known results that assert that a graph satisfies a certain coloringproperty if it admits an appropriate orientation. The most popular results of this typeare the theorem of Galvin [14] based on the kernel method and the theorem of Alon andTarsi [2] using the graph polynomial method. However, one of the oldest result relatingcolorings and orientations is the Gallai–Roy Theorem. It states that the chromaticnumber of a graph is the minimum over all its orientations of the order of a longestdirected path. It seems to have been discovered independently by Gallai [13], Hasse[17], Roy [24] and possibly others. Tuza [26] proved that given an orientation of agraph, where each directed path has length at most k, one can construct in polynomialtime an coloring of this graph using at most k colors. However, many of the knownproofs of such results relating orientations and colorings supply no efficient procedurefor solving the corresponding algorithmic problems. A typical example is the cycle-plus-triangle theorem due to Fleischner and Stiebitz [8].
Theorem 1 (Cycle-Plus-Triangle Theorem) [8] Let n be a positive integer and letG be a 4-regular graph on 3n vertices. Suppose that G has a decomposition into aHamilton cycle and n pairwise vertex disjoint triangles. Then χ(G) = χ
�(G) = 3.
The proof is based on a subtle parity argument showing that, if euo(G) is the numberof Eulerian orientations of a cycle-plus-triangle graph G, then euo(G) ≡ 2 (mod 4).The result then follows from the famous theorem of Alon and Tarsi [2], but the proofsupplies no polynomial time algorithm that produces, for a cycle-plus-triangle graphG and given lists L(v), v ∈ V (G), each of size 3, a (proper) vertex coloring of Gassigning to each vertex a color from its list.
While the method of Alon and Tarsi [2] requires an orientation, where the numbersof odd, respectively even, Eulerian spanning subdigraphs are different, the methodof Galvin [14] requires a kernel-perfect orientation. In addition to that, both methodsrequire an orientation with prescribed out degrees. In this paper, we deal with thecomplexity of the latter orientation problem. Furthermore, we use Galvin’s kernelmethod to prove a Brooks-type result for the list chromatic number of weighted graphs.
1.1 List Colorings of Weighted Graphs
A graph G = (V, E) consists of a finite set V = V (G) of vertices and a set E = E(G)
of 2-element subsets of V , called edges. Thus, our graphs have no loops or multipleedges. An edge {u, v} is usually written as uv or vu.
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Let G be a graph and let X ⊆ V (G). With G[X ] we denote the subgraph of Ginduced by X , further, G−X = G[V (G)\X ]. If X = {x} is a singleton we write G−xrather than G − X . If F ⊆ E(G), then G − F = (V (G), E(G)\F) and if F = {e},we also write G − e. For a vertex v ∈ V (G), let NG(v : X) = {u ∈ X | uv ∈ E(G)},further, let NG(v) = NG(v : V (G)) and NG(X) = ⋃
v∈X NG(v). If f : V (G) → N
is a function, then we define f (X) = ∑v∈X f (v).
The clique number of G is denoted by ω(G). Furthermore, col(G) denotes thecoloring number of G, that is, the smallest integer p ≥ 1 such that G has a vertexorder in which each vertex is preceded by fewer than p of its neighbors. The coloringnumber as a natural upper bound for the chromatic number was introduced by Erdosand Hajnal [6].
A function g : V (G) → N from the vertex set of a graph G to the set of non-negative integers N is called a vertex function of G; the pair (G, g) is then also said tobe a weighted graph. Similarly, a function h : E(G) → N is called an edge functionof G and the pair (G, h) is said to be an edge weighted graph.
We use the following conventions. We say that a weighted graph (G, g) has a certaingraph property (e.g., bipartite or planar) if the underlying graph G has this property. Iff is a vertex function of a graph G, then f can be also considered as a vertex functionfor every subgraph of G. So we denote the restriction of f to a subgraph H also by fand say that f is a vertex function of H . If a vertex function f of a graph G satisfiesf (v) ≥ 1 for every v ∈ V (G), we write f ≥ 1. If (G, g) is a weighted graph and g ≡p, i.e., g(v) = p for every vertex v ∈ V (G), then we write (G, p) rather than (G, g).
Let (G, g) be a weighted graph, let f be a vertex function of G, and let k ≥ 0 bean integer. A list-assignment L of G is a function that assigns to every vertex v of Ga set (list) L(v) of colors (usually each color is a positive integer). We say that L is anf -assignment or a k-assignment if |L(v)| = f (v) for all v ∈ V or |L(v)| = k for allv ∈ V , respectively.
A coloring of (G, g) is a function ϕ that assigns a set of g(v) colors to each vertex v
of G so that ϕ(v)∩ϕ(w) = ∅ whenever vw ∈ E . An L-coloring of (G, g) is a coloringϕ of (G, g) such that ϕ(v) ⊆ L(v) for all v ∈ V . If (G, g) admits an L-coloring, then(G, g) is L-colorable. When L(v) = [1, k] for all v ∈ V (G) (where [1, k] denotesthe set {1, . . . , k} with [1, 0] = ∅), the corresponding terms become a k-coloring andk-colorable, respectively. The weighted graph (G, g) is said to be f -list-colorableif (G, g) is L-colorable for every f -assignment L of G. When f (v) = k for allv ∈ V , the corresponding term becomes k-list-colorable. Note that, in the literature,a coloring of a weighted graph (G, g) is also called a set-coloring of G and if (G, g)
is f -list-colorable it is also said that G is ( f, g)-choosable.The chromatic number of (G, g), denoted by χ(G, g), is the least number k for
which (G, g) is k-colorable. The list-chromatic number or choice number of (G, g),denoted by χ
�(G, g), is the least number k for which G is k-list-colorable. Clearly,
every weighted graph (G, g) satisfies 0 ≤ χ(G, g) ≤ χ�(G, g). Note that if g ≡ 1,
then χ(G, g) becomes the ordinary chromatic number χ(G) of G, and χ�(G, g)
becomes the ordinary list-chromatic number χ�(G) of G.
The study of list-coloring problems for graphs was initiated in the 1970s by Vizing[29] and, independently, by Erdos et al. [7]. Both Vizing and Erdos et al. observed thatbipartite graphs, i.e., graphs with chromatic number at most 2, can have arbitrarily
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large list-chromatic number. The gap between the two parameters χ�(G) and χ(G)
can thus be arbitrarily large. On the other hand, there are several classes of graphs forwhich it is conjectured that χ
�(G) = χ(G). These include line graphs, total graphs,
and claw-free graphs. Classes of graphs for which it is known that χ�(G) = χ(G) are
the chordal graphs (satisfying even a more general equation, cf. [28]), line graphs ofbipartite multigraphs (Galvin [14]), and powers of cycles (Prowse and Woodall [22]).For detailed information on list-colorings, we refer to the surveys by Alon [1], by Tuza[27], by Kratochvíl et al. [19], and by Woodall [31].
It is well known that the coloring problem for weighted graphs can be reduced tothe ordinary coloring problem. With a weighted graph (G, g) we can associate a graphG[g] as follows. For each vertex v ∈ V (G), replace v by a complete graph K v havingg(v) vertices, where all these complete graphs are vertex disjoint, and join K v and K u
by all possible edges whenever uv ∈ E(G). The resulting graph is G[g]. Sometimesthis graph is called an inflation of G. Then, clearly, we have χ(G, g) = χ(G[g]). Onthe other hand, for the list-chromatic number, we only have χ
�(G, g) ≤ χ
�(G[g])
and it is unknown whether equality always holds. To see this, let L be a k-assignmentof G, where k = χ
�(G[g]). Now let L ′ be the k-assignment of G[g] that assigns to
each vertex of K v the same list L(v). Then there is an L ′-coloring ϕ′ of G[g], andso the mapping ϕ that assigns to each vertex v of G the color set ϕ(v) = ϕ′(K v)
is an L-coloring of (G, g). This shows that χ�(G, g) ≤ k, as claimed. Hence, every
weighted graph (G, g) satisfies:
ω(G[g]) ≤ χ(G, g) ≤ χ�(G, g) ≤ χ
�(G[g]) ≤ col(G[g]) ≤ �(G[g]) + 1. (1)
Note that a weighted graph (G, g) satisfies
ω(G[g]) = max{g(X) | X ⊆ V (G) and G[X ] is a complete graph},
and
�(G[g]) = max{g(NG(v) ∪ {v}) | v ∈ V (G)} − 1.
One of the basic results in graph coloring is Brooks’ theorem [4] from 1941, whichasserts that the complete graphs and the odd cycles are the only connected graphs whosechromatic number is larger than their maximum degree. The choosability version ofthis result has been proved by Vizing [29] and, independently, by Erdos et al. [7]. By(1), this result implies that if (G, g) is a connected weighted graph with g ≥ 1, thenχ
�(G, g) ≤ �(G[g]) + 1, where equality holds if and only if G is a complete graph
or g ≡ 1 and G is an odd cycle. In Sect. 3.1 we will prove the following extension ofthis result.
Theorem 2 If (G, g) is a connected weighted graph with g ≥ 1, then
χ�(G, g) ≤ �(G[g]) + 1 − min
u∈V (G)g(u)
unless G is a complete graph or an odd cycle.
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Theorem 2 will be derived from the following result; the proof of this result is alsogiven in Sect. 3.1. As usual, the sum over the empty set is defined to be zero.
Theorem 3 Let (G, g) be a connected weighted graph with g ≥ 1, and let f be avertex function of G such that
f (v) ≥ g(v) +∑
uv∈E(G)
g(u) − min{g(u) | uv ∈ E(G)}
for every vertex v ∈ V (G). Then (G, g) is f -list-colorable unless every block of G isa complete graph or an odd cycle.
For weighted graphs (G, g) with g ≡ 1, the above result was proven by Vizing [29]and by Erdos et al. [7]. For weighted graphs with constant weight function this resultwas proven by Tuza and Voigt [28].
1.2 Orientations of Graphs and Galvin’s Theorem
A directed graph (or digraph) D = (V, E) consists of two finite disjoint sets (ofvertices V = V (D) and edges E = E(D)) together with two maps v+
D : E → Vand v−
D : E → V assigning to every edge e an initial vertex v+D(e) and a terminal
vertex v−D(e) such that v+
D(e) �= v−D(e). The edge e is said to be directed from v+
D(e)to v−
D(e). Thus our digraphs may have several edges between the same two vertices.Such edges are called multiple edges; if they have the same direction, say from u tov, they are parallel. Note that our digraphs have no loops.
A digraph D is an orientation of a graph G if V (D) = V (G) and E(D) = E(G),and if {v+
D(e), v−D(e)} = e for every edge e ∈ E . Intuitively, an orientation of a graph is
a digraph obtained from the graph by directing every edge from one of its end-verticesto the other.
The underlying graph of a digraph D, denoted by G D , is the graph with V (G D) =V (D) and E(G D) = {{v+
D(e), v−D(e)} e ∈ E(D)}. Note that if D is a digraph without
multiple edges, then D is (isomorphic to) an orientation of its underlying graph.Let v be a vertex in a digraph D. The out-neighborhood of v in D, denoted by
N+D (v), is the set of all vertices u in D such that D contains an edge directed from
v to u. The out-degree d+D(v) is the number of edges in D with initial vertex v. The
maximum out-degree is denoted by �+(D).A set X ⊆ V (D) is called an independent set or a clique of D if X is an independent
set or a clique of the underlying graph G D of D, respectively. A kernel in the digraphD is an independent set S of D such that for every vertex u ∈ V (D)\S there is an edgein D directed from u to some vertex v ∈ S. A digraph D is kernel-perfect if everyinduced subdigraph of D has a kernel. The subdigraph of D induced by X is denotedby D[X ].
The following result due to Galvin [14] became a very useful tool for solvinglist coloring problems. The unweighted case of this result was obtained by Bondy,Boppona and Siegel (unpublished, see [14]).
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Fig. 1 A kernel perfectsuper-orientation of K −
4
Theorem 4 [14] Let (G, g) be a weighted graph and let f be a vertex function of G.If there exists a kernel perfect digraph D such that G = G D and
f (v) ≥ g(v) +∑
u∈N+D (v)
g(u) (2)
whenever g(v) ≥ 1, then (G, g) is f -list-colorable.
Note that in Galvin’s theorem it is not required that D is an orientation of G. Onthe other hand, parallel edges are not helpful in order to construct a kernel perfectdigraph D satisfying G = G D and (2). A digraph D without parallel edges satisfyingG = G D is sometimes called a super-orientation of G. Several authors do not distin-guish between orientations and super-orientations. If there is a super-orientation of Gsatisfying (2), then there is also an orientation of G satisfying this condition.
For the graph G = K −4 , there exists a kernel perfect super-orientation D (see
Fig. 1) such that every vertex v of G satisfies d+D(v) ≤ dG(v)− 1, but there is no such
orientation of G.Clearly, a kernel in a digraph is a maximal independent set, but the converse state-
ment is not true and kernels may even fail to exists. For example, an odd directed cyclehas no kernel. In 1973 Chvátal [5] proved that it is NP-complete to decide whether adigraph has a kernel, or not. On the other hand, Richardson [23] proved in 1953 thefollowing positive result.
Theorem 5 [23] Every digraph having no odd directed cycle has a kernel and istherefore kernel-perfect.
Obviously, every digraph D such that the graph G D is bipartite contains no odddirected cycle. This simple observation together with Galvin’s result implies the fol-lowing.
Corollary 6 Let (G, g) be a bipartite weighted graph and let f be a vertex functionof G. If there exists an orientation D of G such that
f (v) ≥ g(v) +∑
u∈N+D (v)
g(u)
whenever g(v) ≥ 1, then (G, g) is f -list-colorable.
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By a weighted digraph we mean a pair (D, g) such that D is a digraph and g :V (D) → N is a vertex function of D. For such a weighted digraph (D, g), weintroduce the following notation. The weighted out-degree of a vertex v in (D, g) isdefined by
d+D,g(v) =
∑
u∈N+D (v)
g(u).
Furthermore, let �+(D, g) denote the maximum weighted out-degree in (D, g). If Dis a directed graph without parallel edges, then d+
D,1(v) = d+D(v) and �+(D, 1) =
�+(D).Let (G, g) be a weighted graph, and let f be a vertex function of G. A digraph D is
said to be an f -orientation of (G, g) if D is an orientation of G such that every vertexv ∈ V (G) satisfies d+
D,g(v) ≤ f (v). An f -orientation of (G, 1) is also said to be anf -orientation of G.
If the weighted graph (G, g) is bipartite, then Corollary 6 says that (G, g) is f -list-colorable provided that there exists an f ′-orientation of (G, g) for the vertex functionf ′ = f − g. This motivated us to investigate the complexity of the WeightOrient
problem of deciding if, for a given weighted graph (G, g) and a given vertex functionf of G, there exists an f -orientation of (G, g). As proved in Sect. 2.1, the problemWeightOrient is NP-complete. The BP2- WeightOrient problem is the restrictionof WeightOrient to the class of bipartite planar weighted graphs (G, g) with �(G) ≤3 and to vertex functions f of G such that the range of f as well as of g is the set{1, 2}. In Sect. 2.1 we shall prove the following complexity result.
Theorem 7 BP2- WeightOrient is NP-complete.
If we only allow weighted graphs (G, g) with a constant function g, but witharbitrary vertex functions f , we obtain another variant referred to as UnWeightO-
rient. The following result proved in Sect. 2.2 implies that the decision problemUnWeightOrient belongs to P.
Theorem 8 There exists a polynomial time algorithm whose input is an arbitrarygraph G and an arbitrary vertex function f of G and whose output is either an f -orientation D of G or a certificate showing that such an orientation does not exist.
2 Orientations of Weighted Graphs
2.1 The Weighted Case with Variable Out-Degrees
The goal of this section is to prove Theorem 7 saying that BP2- WeightOrient isNP-complete. Furthermore, we show that the analogue orientation problem for edgeweighted graphs is NP-complete, too.
Proof of Theorem 7 The canonical problem for BP2- WeightOrient is 3- SAT.For an instance φ = C1 ∧ · · · ∧ Cn of 3- SAT with clauses C1, . . . , Cn and variablesx1, . . . , xm , we define the bipartite graph Hφ with color classes (X, Y ) as follows. The
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vertex set X = {x1, . . . , xm} represents the variables occurring in φ, the vertex setY = {C1, . . . , Cn} represents the clauses occurring in φ, and a pair xi C j belongs tothe edge set of Hφ if the variable xi occurs in the clause C j as a positive or negativeliteral. Then φ is said to be planar formula if the graph Hφ is planar.
Let PV3- SAT denote the restriction of 3- SAT to planar formulae in conjunctivenormal form, where each variable occurs in exactly three clauses, and each clause hasat most three literals. The NP-completeness of PV3- SAT was proven in [21].
Let PVC- SAT denote the restriction of 3- SAT to planar formulae φ such that
(A) each clause of φ is the disjunction of two or three literals,(B) each variable of φ occurs in at most three clauses, and(C) each variable occurs both as a positive literal and as a negative literal, but not
together in a same clause of φ.
We claim that PVC- SAT is NP-complete, too. To see this, let φ be an instance ofPV3- SAT. If φ has a clause consisting of exactly one literal �, then delete all clausecontaining �, and delete the negation ¬� in all clauses containing ¬�. By repeatedapplication of this reduction, we obtain a planar formula φ′ satisfying (A) and (B),where φ is satisfiable if and only if φ′ is satisfiable. If a variable x occurs in φ′ onlyas a positive literal (or only as a negative literal), then delete all clauses containing x(respectively, ¬x). Furthermore, delete all clause containing both x and ¬x . Again,by repeated application of this reduction we obtain a planar formula φ′′ satisfying (A),(B), and (C), where φ′ is satisfiable if and only if φ′′ is satisfiable. This proves theclaim that PVC- SAT is NP-complete.
For each instance φ of PVC- SAT we construct a graph Gφ and specify two vertexfunctions f, g : V (Gφ) → {1, 2} with the following property:
(a) (Gφ, g) has an f -orientation iff φ is a positive instance of PVC- SAT.
Let φ = C1 ∧ · · · ∧ Cn be an instance of PVC- SAT, where the variables arex1, . . . , xm , and let H = Hφ . Since φ is a planar formula, the bipartite graph H withcolor classes X = {x1, . . . , xm} and Y = {C1, . . . , Cn} is planar. Recall that xi C j isan edge of H if and only if xi occurs in C j as a positive or as a negative literal. Thusthe neighborhood of each variable xi ∈ X in H is divided into two sets N+
i and N−i ,
where the former set consists of all clauses C j ∈ Y such that xi occurs in C j and thelatter set consists of all clauses C j ∈ Y such that ¬xi occurs in C j . On the one hand,we have �(H) ≤ 3 (by (A) and (B)). On the other hand, we deduce that N+
i | ≤ 3 (by (B) and (C)).Let the elements of Z = {x+
1 , x−1 , . . . , x+
m , x−m } represents the positive and negative
literals depending on whether the superscript is ’+’ or ’−’.We now define the graph G = Gφ as follows. The vertex set of G is V (G) =
X ∪ Y ∪ Z . The edge set of G consists of two types of edges. For each variable xi , wehave the two edges xi x+
i and xi x−i , and for each clause C j and each literal � occurring
in C j , we have the edge C j x+i if � = xi or the edge C j x−
i if � = ¬xi . Note that Gis obtained from H by applying to each vertex xi the following local operation. Since{N+
i , N−i } is a partition of NH (xi ) and 2 ≤ |N+
i | + |N−i | ≤ 3, we have |N+
i | = 1 or|N−
i | = 1. If N+i consists of exactly one clause C j , then rename the vertex xi in x−
iand replace the edge x−
i C j by the path (x−i , xi , x+
i , C j ). If otherwise N−i consists of
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Fig. 2 An instance φ of PVC- SAT and the graphs Hφ and Gφ
exactly one clause C j , then rename the vertex xi in x+i and replace the edge x+
i C j
by the path (x+i , xi , x−
i , C j ). The resulting graph is G. Since H is a bipartite planargraph with �(H) ≤ 3, we deduce that G is a bipartite planar graph with �(G) ≤ 3,too (Fig. 2).
The two vertex functions f, g : V (G) → {1, 2} are specified as follows: g(v) =2, f (v) = 1 if v ∈ X , g(v) = 1, f (v) = 2 if v ∈ Z and g(v) = 1, f (v) = |C | − 1if v = C ∈ Y , where |C | is the number of literals occurring in C . Hence (G, g) andf is an instance of BP2- WeightOrient. To complete the proof, it suffices to showthat (a) holds.
For the proof of the “if” part, assume that φ is a positive instance of 3- SAT. Thenthere is a truth assignment ν : {x1, . . . , xm} → {0, 1} such that the truth value of φ
is 1 (=true). Now, we orient the edges of G as follows. The only out-neighbor of thevariable xi is the literal in {x+
i , x−i } whose truth value is 1. The out-neighbors of a
clause C j are those literals of C j whose truth value is 0. Then, for a variable xi , wehave d+
D,g(xi ) = d+D(xi ) = 1 = f (xi ). Since the truth value of φ is 1, for each clause
C j , we have d+D,g(C j ) = d+
D(C j ) ≤ |C j |−1 = f (C j ). Now, consider a vertex � ∈ Z .This vertex corresponds to a literal. If the truth value of this literal is 1, then � has atmost two out-neighbors and both are in Y . Then we have d+
D,g(�) ≤ 2 = f (�). If thetruth value of this literal is 0, then � has exactly one out-neighbor x belonging to Xand, therefore, d+
D,g(�) = 2 = f (�). This shows that D is an f -orientation of (G, g).For the proof of the “only if” part, assume that D is an f -orientation of (G, g). Then
we have d+D,g(x) ≤ 1 for all x ∈ X , d+
D,g(z) ≤ 2 for all z ∈ Z and d+D,g(C) ≤ f (C) =
|C |−1 for all C ∈ Y . This implies, in particular, that each variable xi ∈ X has at mostone out-neighbor in D. We now define a truth assignment ν : {x1, . . . , xm} → {0, 1}as follows. If d+
D(xi ) = 1, then let ν(xi ) = 1 if the out-neighbor of xi is x+i and let
ν(xi ) = 0 otherwise. If d+D(xi ) = 0, then let ν(xi ) = 1. We now claim that the truth
value of φ is 1. To prove the claim, let C j be one of the clauses of φ. Since g(�) = 1for all neighbors � of C j in G, we have d+
D(C j ) = d+D,g(C j ) ≤ f (C j ) = |C j | − 1.
This implies that in D there is an edge directed from a vertex � ∈ Z to C j . Note that
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� ∈ {x+i , x−
i } for some i ∈ {1, . . . , m} and the corresponding literal occurs in C j .Since d+
D,g(�) ≤ f (�) = 2, we then conclude that there is an edge in D directed fromxi to �. This implies that the truth value of � and hence of C j is 1. So φ is a positiveinstance of 3- SAT as required. This completes the proof. ��
The decision problem WeightOrient has an obvious edge version, calledEdgeWeightOrient. For a digraph D and a vertex v of D, let E+
D(v) denote theset of all edges e ∈ E(G) such that v+
D(e) = v. The input of EdgeWeightOrient
is an edge weighted graph (G, h) and a vertex function f of G and the question iswhether there exists an orientation D of G such that each vertex v ∈ V (G) satisfies
f (v) ≥∑
e∈E+D(v)
h(e). (3)
There are several examples in graph theory showing that the edge version of adifficult vertex problem becomes much easier. However, the next result shows thatthis is not the case for the vertex orientation problem.
Theorem 9 EdgeWeightOrient is NP-complete even if we restrict the input tobipartite planar edge weighted graphs (G, h) of maximum degree at most three andwith h(e) ∈ {1, 2} and f (v) ∈ {1, 2}.Proof The proof is very similar to the former proof. In particular, we use again thesame version of 3- SAT. Let φ = C1 ∧ · · · ∧ Cn be an instance of PVC- SAT withvariables x1, . . . , xm . Then, let G = Gφ be the same graph as in the proof of Theorem7. Next, let h be the edge function of G, where h(e) = 2 if e is an edge of G − Y andh(e) = 1 otherwise. Eventually, let f (v) = 2 if v ∈ X ∪ Z and f (v) = |C | − 1 ifv = C ∈ Y . To complete the proof, it suffices to show the following property:
(b) There is an orientation D of Gφ such that each vertex v satisfies the inequality(3) iff φ is a positive instance of PVC- SAT.
Since the proof of (b) is similar to the proof of property (a) in Theorem 7, the detailsare left to the reader. ��
2.2 The Unweighted Case with Variable Out-Degrees
In this section we deal with the orientation problem for graphs and give a proof of The-orem 8. Given a graph G and a vertex function f of G, we want to find an f -orientationof G, that is, an orientation D of G such that d+
D(v) ≤ f (v) for every vertex v ∈ V (G),provided that such an orientation exists. In the mid 1960s Hakimi [15] investigatedproperties of the degree sequence of directed graphs and presented at the end of hispaper a necessary and sufficient condition for the existence of an f -orientation of agraph G formulated in a slightly different way. The rather long proof he gave is byinduction on the order of G. An elegant proof from first principles was given by Frankand Gyarfás [9] (see also [10]). They also remarked that this orientation problem canbe formulated as a network-flow problem and that the characterization follows fromwell-known results. A proof of this characterization based on Hall’s theorem [16] is
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included in the famous paper by Alon and Tarsi [2]. For the reader’s conveniencewe will give both short proofs here, so the reader can see that Theorem 8 is a directconsequence of these proofs and the well-known fact that both the maximum flowproblem and the matching problem for bipartite graphs can be solved in polynomialtime. In particular, there exists a polynomial time algorithm that computes for anygiven bipartite graph G with color classes (A, B) either a matching of size |A| or aset S ⊆ A with |NG(S)| < |S|.Theorem 10 [15] Let G be a graph and let f be a vertex function of G. Then G hasan f -orientation if and only if |E(G[X ])| ≤ ∑
v∈X f (v) for every X ⊆ V (G).
Proof The “only if” part is evident. For the proof of the “if” part, we have to showthat there is an appropriate orientation of G.
Method 1: In an orientation D of G call a vertex v bad if d+D(v) > f (v). Choose
an orientation D where the “badness”∑
(d+D(v) − f (v) : v bad) is minimal. If there
is no bad vertex, we are done. Otherwise, let s be a bad vertex and let X be the set ofall vertices v such that there is a directed path in D from s to v. Then s ∈ X and if eis an edge of D with v+
D(e) ∈ X then v−D(e) ∈ X . Consequently, X contains a vertex
t with d+D(t) ≤ f (t) − 1, since otherwise |E(G[X ])| = ∑
v∈X d+D(v) >
∑v∈X f (v),
contradicting the assumption. Reorienting the edges of a directed path from s to t inD results in an orientation with a smaller badness.
Method 2: First, we construct an auxiliary bipartite graph F with color classes(A, B) as follows. The set A consists of the edges of G and the set B consists of f (v)
copies for each vertex v of G. An edge e = uv is then joined in F by an edge toeach copy of u as well as to each copy of v. We claim that F contains a matchingof size |A|. To see this, we apply Hall’s condition. For an arbitrary edge set E ′ ⊆ A,let H be the subgraph of G such that V (H) is the set of all end-vertices of edges inE ′ and E(H) = E ′. For the neighborhood of E ′ in the bipartite graph F , we clearlyhave |NF (E ′)| = ∑
v∈V (H) f (v). By assumption, we also have |E ′| = |E(H)| ≤∑v∈V (H) f (v). Hence Hall’s condition |NF (E ′)| ≥ |E ′| holds and, therefore, there
exists an matching of size |A| in F . We can now orient each edge of G from the vertexto which it is matched in F . Clearly, this results in an f -orientation of G. ��
3 Colorings of Weighted Graphs
3.1 Proof of Brooks’s Theorem for Weighted Graphs
The goal of this section is to prove Theorems 2 and 3. A connected graph all of whoseblocks are complete graphs and/or odd cycles is called a Gallai tree; this class ofgraphs was introduced by Gallai [12]. Theorem 3 is an immediate consequence ofTheorem 4 and the following result.
Theorem 11 A connected graph G has a kernel perfect super-orientation D such thatevery vertex v of G satisfies d+
D(v) ≤ dG(v) − 1, provided that G is not a Gallai tree.
Proof Suppose that G is a connected graph, but not a Gallai tree. By a well knownstructural result of Erdos et al. [7], G contains an even cycle C with at most one
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chord as an induced subgraph. If we contract C to a single vertex wC , we obtain aconnected graph G∗. Now, using a spanning tree of G∗, it is easy to construct an acyclicorientation of G∗ such that each vertex of G∗ except wC has in-degree at least one.Let D∗ be the corresponding orientation of G − E(G[V (C)]). We extend D∗ to ansuper-orientation D of G by orienting the edges of the cycle of C in a cyclic order and,moreover, by replacing the chord uv of C , if such a chord exists, by two directed edges,one directed from u to v and one directed from v to u. Clearly, d+
D(v) ≤ dG(v) − 1for every vertex v of G.
The proof is completed by showing that D is kernel perfect. To this end, let D′ bean induced subdigraph of D with n ≥ 1 vertices. Then we show by induction on nthat D′ has a kernel. If n = 1, this is evident.
Now, assume that n ≥ 2. If D′ has a vertex v with out-degree zero, then by induction,there is a kernel K of D′ − (N−
D′(v)∪{v}) and K ∪{v} is a kernel of D′, where N−D′(v)
denotes the set of all vertices u ∈ V (D′) such that there is an edge in D′ directed fromu to v. Otherwise, D′ is an induced subdigraph of D[V (C)] and we argue as follows.If C has no chord, then D′ has no directed odd cycle and Theorem 5 implies that D′has a kernel. Now, assume that C has a chord uv. If D′ = D[V (C)] then take K asthe largest independent set. Since C is an even cycle with at most one chord, K is akernel of D′. If D′ is a proper induced subgraph of D[V (C)], then let e1, e2 be thetwo edges of D such that {v+
D(ei ), v−D(ei )} = {u, v} for i = 1, 2. Then D1 = D′ − e1
or D2 = D′ − e2 does not contain an directed odd cycle. Hence Theorem 5 impliesthat either D1 or D2 has a kernel K . Clearly, K remains a kernel of the digraph D′.
��Our proof of Theorem 3 resembles the original proof by Erdos et al. [7] for their
result saying that every connected graph G which is not a Gallai tree is dG -list-colorable, where dG is the degree function of G. However, in [7] the constructionof an appropriate list-coloring is not based on Galvin’s kernel argument, but a moredirect argument avoiding a super-orientation of C . While it is easy to show that an evencycle C with at most one chord is dC -list-colorable, it seems to be difficult to prove thecorresponding statement for the weighted graph (C, g) without using Galvin’s kernelmethod.
Our proof of Theorem 3 yields a polynomial time algorithm that finds for everyf -list assignment L of (G, g) an L-coloring of (G, g) provided that G is a connectedgraph which is not a Gallai tree and
f (v) ≥ g(NG(v) ∪ {v}) − min{g(u) | u ∈ NG(v)}
for every vertex v of G. The proof of Theorem 11 tells us how the required kernelscan be constructed in polynomial time. In general this need not be the case. Even ifwe know that a digraph D is kernel-perfect, it might be difficult to construct a kernelfor an induced subdigraph.
Note that the super-orientation D of G constructed in the proof of Theorem 11satisfies the Alon-Tarsi condition, since either D has no Eulerian spanning subdigraphwith an odd number of edges, but at least one such subdigraph with an even number ofedges, or D has three Eulerian spanning subdigraphs with an even number of edges,
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but only two such subdigraph with an odd number of edges. Hence also the result ofAlon and Tarsi [2] implies that G is dG-list-colorable if no component of G is a Gallaitree. That such an orientation exists provided that no component is a Gallai tree wasfirst discovered by Hladký et al. [18]. However, while Galvin’s kernel method alsoapplies to list-colorings of weighted graphs, the method of Alon and Tarsi does not. Tosee this, consider the graph G = K4 − xy and let u, v be the two vertices of G − x − y.Let D be the digraph with V (D) = V (G), where the edges of the cycle (x, u, y, v) areoriented in cyclic order and the edge uv is directed from v to u. Let g, f be the vertexfunctions with g(x) = 1, g(u) = g(y) = g(v) = 2, f (x) = 3, f (u) = f (y) = 4,and f (v) = 5. Then D satisfies the Alon–Tarsi condition and f satisfies the inequality(2) for every vertex of G, but it is easy to find an uncolorable list assignment provingthat (G, g) is not f -list-colorable.
If G is a connected graph and w is any vertex of G, then there is an acyclic orientationD of G such that d+
D(v) ≤ dG(v) − 1 for every vertex v of G except w. Since anacyclic orientation is kernel perfect, Theorem 4 then implies the following result.
Proposition 12 Let (G, g) be a connected weighted graph with g ≥ 1, let w be avertex of G, and let f be a vertex function of G such that
f (v) ≥ g(v) +∑
uv∈E(G)
g(u) − min{g(u) | uv ∈ E(G)}
for every vertex v ∈ V (G)\{w} and
f (w) ≥ g(w) +∑
uw∈E(G)
g(u).
Then (G, g) is f -list-colorable.
Combining Proposition 12 and Theorem 3, we derive a proof of Theorem 2.Proof of Theorem 2 Let (G, g) be a connected weighted graph with g ≥ 1 and let
L be an arbitrary k-assignment of G, where
k = �(G[g]) + 1 − minu∈V (G)
g(u).
We have to show that (G, g) is L-colorable, provided that G is neither a complete graphnor an odd cycle. If G is not a Gallai tree, this follows immediately from Theorem 3.If G is a Gallai tree, then we argue as follows. Since G is neither a complete graphnor an odd cycle, G consists of more than one block. This implies that there is a blockB of G containing exactly one cut vertex of G. Let w denote this cut vertex and,moreover, let w1, w2, . . . , wr be the neighbors of w in B and let u1, u2, . . . , us be theneighbors of w in G − V (B). Note that we have r, s ≥ 1. By Proposition 12, thereexists an L-coloring ϕ′ of (G − V (B), g). Let L ′ be the list-assignment of B suchthat L ′(v) = L(v) if v ∈ V (B − w) and L ′(w) = L(w)\(ϕ′(u1) ∪ · · · ∪ ϕ′(us)).Our aim is to show that (B, g) is L ′-colorable. Clearly, this would imply that (G, g)
is L-colorable. Let m = minu∈V (G) g(u). Then we have
and, therefore, |L ′(w1)\L ′(w)| ≥ g(u1) ≥ m. To construct an L ′-coloring of (B, g),we first color the vertex w1 by a set A of g(w1) colors chosen from the list L ′(w1). Since|L ′(w1)\L ′(w)| ≥ m, this can be done in such a way that at most g(w1) − m colorsof A are contained in L ′(w). Now, consider the list-assignment L1 of the connectedgraph B1 = B − w1 such that L1(v) = L ′(v) − A if vw1 ∈ E(B) and L1(v) = L ′(v)
Since B1 is connected, Proposition 12 then implies that there is an L1-coloring of(B1, g) and, therefore, also an L ′-coloring of (B, g). This completes the proof. ��
3.2 Special Graph Classes
Recall that a graph G is perfect if every induced subgraph H of G satisfies χ(H) =ω(H). By a result of Lovász [20], the inflation graph of a perfect graph is perfect, too.Clearly, there is a one-to-one correspondence between the weight functions g withrange {0, 1} of a graph G and the induced subgraphs of G. Hence, a graph G satisfiesχ(G, g) = ω(G[g]) for every weight function g if and only if G is perfect. Let us calla graph G strong list-perfect if χ
�(G, g) = ω(G[g]) for every weight function g of
G. We know two classes of strong list-perfect graphs, namely, line graphs of bipartitemultigraphs and chordal graphs.
Let G be the line graph of a bipartite multigraph and let g be an arbitrary weightfunction of G. Then the inflation G[g] is also the line graph of a bipartite multigraph.Hence Galvin’s result [14] implies that χ
�(G[g]) = χ(G[g]) = ω(G[g]). By (1), this
implies that G is strong list-perfect.Let G be a chordal graph and let g be an arbitrary weight function of G. Then the
inflation G[g] is also a chordal graph. Clearly, chordal graphs are perfect. That thelist-chromatic number and the chromatic number of a chordal graph are equal was
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first observed by Tesman [25]. By (1), it then follows that G is strong list-perfect.Fulkerson and Gross [11] proved that a chordal graph G has a simplicial ordering,that means an order v1, v2, . . . , vn of its vertices such that, for every 1 ≤ i ≤ n, therestricted neighborhood NG(vi : {vi+1, vi+2, . . . , vn}) induces a complete graph inG. This simplicial ordering of G can be used to find an acyclic orientation D of G suchthat N+
D (v) is a clique of G. Since an acyclic orientation is kernel-perfect, Theorem11 implies the following result.
Proposition 13 Let (G, g) be a chordal weighted graph and, for every vertex v ∈V (G), let Xv be a clique of G containing v such that g(Xv) is maximum. Furthermore,let f be a vertex function of G such that f (v) ≥ g(Xv) for every vertex v ∈ V (G).Then (G, g) is f -list-colorable.
Another family of weighted graphs for which the list-chromatic number equals thechromatic number are the weighted odd cycles. As proved by Woodall [30] everyweighted odd cycle (C, g) satisfies
χ�(C[g]) = χ
�(C, g) = χ(C, g) = max
{
ω(C[g]), 2g(C)
|C | − 1
}
.
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