Graph colorings with local constraints — A survey ∗

Zsolt Tuza y

Latest update : September 8, 1997

Abstract Contents

We survey the literature on those variants of the chro- 0 Introduction 2 matic number problem where not only a proper coloring 0.1 Standard definitions ...... 2 has to be found (i.e., adjacent vertices must not receive 0.2 Notation for vertex colorings ...... 3 the same color) but some further local restrictions are im- 0.3 Some variations ...... 3 posed on the color assignment. Mostly, the list colorings 0.4 Small uncolorable graphs ...... 3 and the precoloring extensions are considered. In one of the most general formulations, a graph 1 General results 4 G =(V,E), sets L(v) of admissible colors, and natural 1.1 Equivalent formulations ...... 4 numbers c for the vertices v ∈ V are given, and the ques- v 1.2 Complete bipartite graphs and the con- tion is whether there can be chosen a subset C(v) ⊆ L(v) struction of Haj´os ...... 4 of cardinality cv for each vertex in such a way that the sets 1.3 Typical behavior of the choice number .. 5 C(v),C(v ) are disjoint for each pair v, v of adjacent ver- 1.4 Unions of graphs and the (am, bm)- tices. The particular case of constant |L(v)| with cv =1 for all v ∈ V leads to the concept of choice number,a conjecture ...... 6 graph parameter showing unexpectedly different behavior 1.5 Graphs and their complements ...... 7 compared to the chromatic number, despite these two in- variants have nearly the same value for almost all graphs. 2 Vertex degrees 7 To illustrate typical techniques, some of the proofs are 2.1 The theorems of Brooks and Gallai .... 7 sketched. 2.2 Lower bounds on the choice number ... 8 2.3 Graph polynomials ...... 9 2.4 Orientations and Eulerian subdigraphs .. 10

Keywords : graph coloring, list coloring, choice number, 3 Comparisons of coloring parameters 11 precoloring extension, complexity of algorithms, chro- 3.1 Planar graphs ...... 11 matic number 3.2 Graphs with equal chromatic and choice number ...... 12 AMS Subject Classification : 05–02, 05C15 (pri- 3.3 Edge and total colorings ...... 13 mary) ; 68R10 (secondary) 3.4 Choice ratio and fractional chromatic number 15 3.5 The chromatic polynomial ...... 16

4 Algorithmic complexity 17 4.1 Precoloring extension ...... 17 4.2 Good characterizations ...... 19 c Copyright 4.3 List colorings ...... 19 4.4 Choosability ...... 22 This article appears in: 4.5 Graph coloring games ...... 22 Discussiones Mathematicae – Graph Theory, Vol. 17, No. 2 (1997), 161–228. References 23

∗ Research supported in part by the Hungarian National Re- search Fund through grant OTKA T–016416. † Computer and Automation Institute, Hungarian Academy of Sciences, H–1111 Budapest, Kende u. 13–17, Hungary. E-mail : [email protected] ; URL : http://www.sztaki.hu/∼tuza/

1 0 Introduction ences ; see [89, 174] and the surveys [154, 155]. Precol- oring extension also has some consequences on the non- The key concept of this survey, list coloring, was intro- approximability of some scheduling problems [22]. More- duced in the second half of the 1970s, in two papers, over, edge colorings of complete bipartite (and also of by Vizing [190] and independently by Erd˝os, Rubin and complete) graphs have equivalent interpretations in terms Taylor [62]. Despite the subject offers a large number of Latin squares and rectangles. The extendability of par- of challenging problems, some of which appeared already tial Latin squares has been studied extensively ; we refer in [62], the vertex list colorings remained almost forgotten to the survey [10] and the more recent paper [11] for ref- for about a decade. The field started to flourish around erences in this part of the literature. 1990, and has attracted an increasing attention since then. From the theoretical point of view, Vizing introduced Most of the early questions have been answered, and new list colorings with the intention to study total colorings, directions have been initiated. But one of the innocent- while Erd˝os, Rubin and Taylor took their motivation from looking problems raised in [62] (Problem 1.5 below) is Dinitz’s conjecture on n × n matrices. Last but not least, still open, and in the particular cases for which affirma- the idea of extending a partial coloring to a larger one tive answers have been proved, we are still rather far from is a natural approach in various contexts where graph a general solution. colorings are constructed sequentially. The systematic study of precoloring extensions was initiated about a decade after [62], in the paper by Bir´o, Related problems. At the end of this informal intro- Hujter and Tuza [18]. Some of its particular cases (mostly duction, let us say a few words also about three topics in connection with edge colorings) appeared earlier in the that will not be considered here, despite they might have works of Burr [40], Marcotte and Seymour [145], and, fitted nicely in the context. First, we shall not deal with using a different terminology, in several papers on Latin problems in which some forbidding condition (e.g., the ex- squares. clusion of ‘ being monochromatic ’) is extended from ad- In this paper we summarize what is known so far on jacent vertices to vertex pairs at larger distance apart. these problems and in their ‘ close neighborhood.’ Sur- These ‘ distance colorings ’ lead to interesting questions veying this part of the literature, not only the strongest and results, but usually may be viewed as colorings on results but also much of the history is presented. Some the kth powers of graphs, and so they are less ‘ restricted ’ typical techniques are illustrated by sketches of proofs. than the concepts discussed here. Second, in a more gen- P Several open problems are mentioned, too. eral setting, the ‘ -chromatic number ’ of a graph can be defined with respect to any hereditary property P.This We have to mention at this point that the class of hy- concept is discussed in detail in the paper [33], therefore pergraphs seems to offer a big unexplored area with many we shall only mention a couple of related references at interesting results to be discovered. And, in this con- some points. Last but not least, we do not consider here text as well, the intensively studied symmetric structures ‘ rankings,’ i.e., vertex (edge) colorings with positive in- (finite geometries, Steiner systems, balanced incomplete tegers in such a way that each monochromatic pair of block designs) may deserve more attention. vertices (edges) is completely separated by the vertices There are at least two previous works to be cited for (edges) of greater colors. A large part of the literature general reference on list colorings. The paper of Alon [4] on rankings can be traced back from the relatively recent surveys the early results, presents some of the important papers [125] and [21]. The unpublished manuscript [184] methods, and also contains several new theorems. More- surveys many problems ; we hope to polish this prelimi- over, many aspects of list colorings, with lots of inter- nary version for publication reasonably soon. esting historical remarks and informative comments, are discussed in various subsections of the excellent book by Jensen and Toft [111]. 0.1 Standard definitions

Applications. Before giving the formal definitions, let A graph (meant to be undirected, without loops and mul- us mention that both List Coloring and Precoloring tiple edges) or multigraph (undirected, without loops) Extension are well motivated, providing natural inter- will usually be written in the form G =(V,E), where pretations for various kinds of scheduling problems ; see, V = V (G)andE = E(G) denote the set of vertices and e.g., [18, 19, 22]. As a matter of fact, the starting point of edges, respectively. The complement of G will be denoted the investigations on precoloring extension was the obser- by G,thedegree of vertex v by d(v)ordH (v) when the vation that, on interval graphs, it provides an equivalent particular graph H in which it is considered has to be formulation of a practical problem where flights have to emphasized, and the maximum degree of G by Δ(G)or be assigned to a given number of airplanes according to Δ. The cardinality |V | of the vertex set is called the order the schedule of a timetable, under the additional condi- of G, and usually will be denoted by n. The parameters tion that the fixed schedule of maintenances (prescribed α(G)andω(G)denotetheindependence or stability num- for each airplane) must not be changed. Further applica- ber and the clique number, respectively (i.e., the largest tions include issues in VLSI theory. The problem of T - cardinality of a subset Y ⊆ V consisting of mutually non- Colorings has important practical motivation as well, adjacent resp. adjacent vertices). Standard notation is from the area of frequency assignments to avoid interfer- applied for particular types of graphs, too, including Kn

2 (complete graph with n vertices), Kp,q (complete bipar- that extends ϕW . That is, a color should be assigned to tite graph with vertex classes of respective cardinalities p each precolorless vertex vi ∈ V \ W from the list Li := and q), Pn (path of length n − 1), Cn (cycle of length n), {1,...,k} (identical lists for the entire V \ W ) while the Sn (star of degree n − 1). Terminology not defined here colors Lj := {ϕW (vj )} of the precolored vertices vj ∈ W for particular classes of graphs and basic concepts can be are unchangeable. The parameter k is termed the color found e.g. in [15, 29, 72, 91, 142]. bound. A proper vertex / edge / total coloring is a mapping Finally, the coloring number of G, denoted col(G), is ϕ from the set V / E / V ∪ E into the set IN of natural defined as the largest integer k such that G has a subgraph numbers, such that the first / the second / all the three of minimum degree k − 1. Equivalently, col(G)isthe conditions below are satisfied : smallest k such that G is ‘ (k−1)-degenerate.’ As a trivial first remark, let us note that if v has more colors in its list •  ∈ n ϕ(v) = ϕ(v ) for all vertex pairs v, v V with than the number of its neighbors, then G is list colorable vv ∈ E, if and only if so is G − vn. In this way, the inequalities • ϕ(e) = ϕ(e)foreachpaire, e ∈ E of edges sharing χ(G) ≤ χ (G) ≤ col(G) ≤ Δ(G)+1 avertex, are valid for every graph G. • ϕ(v) = ϕ(e) for all v ∈ V and all e ∈ E with v ∈ e. Throughout the paper, the expressions ‘ coloring ’ and 0.3 Some variations ‘ proper coloring ’ will be used as synonyms, except in the few paragraphs where the ‘ T -colorings ’ are considered Beside the concepts introduced above, at some points we (see the definition in Subsection 0.3). We shall mostly shall mention results on the following variants, too. deal with vertex colorings ; the only exceptions are some (f,g)-choosability. A more general setting for k- complexity issues (in Section 4) and the material pre- choosability is as follows. Let f and g be two functions sented in Section 3.3. from the same domain V into IN, with f(vi) ≥ g(vi)for all 1 ≤ i ≤ n. The graph G is said to be (vertex-) (f,g)- 0.2 Notation for vertex colorings choosable if, for every list assignment L with |Li| = f(vi) for all i, there can be chosen subsets Si ⊆ Li of cardi- Assuming that the vertex set is V = {v1,...,vn}, Li will nality |Si| = g(vi), such that Si ∩ Sj = ∅ holds for every denote the list (= set of admissible colors) associated with edge vivj ∈ E. The constant functions are of particu- vi. For the union of the lists, we use the notation lar interest ; the case g ≡ 1istermedf-choosable, while f ≡ k and g ≡  with k,  ∈ IN fixed will be referred to as IL : = L1 ∪···∪Ln . (k, )-choosable. These concepts extend to edge and total We also denote colorings in the natural way. (p, q, r)-choosability. This type of list colorings is ob- L := (L1,...,Ln) , tained from the previous one by taking constant functions the (ordered) n-tuple of lists. A mapping ϕ : V → IL i s f ≡ p and g ≡ q, and assuming that |Li ∪ Lj|≥p + r a(vertex)list coloring,oranL-coloring,ifϕ is a proper whenever vi and vj are adjacent. To exclude trivial un- coloring and ϕ(vi) ∈ Li holds for all 1 ≤ i ≤ n.(Insome colorability, it is assumed that p ≥ q and p + r ≥ 2q. papers, the set of forbidden colors is given instead of the List T -colorings. Given a set T ⊂ IN ∪{0}, a (vertex) admissible ones. Those sets may be viewed as comple- T -coloring of G =(V,E) is a mapping ϕ : V → IN s u c h ments of the L with respect to IL.) i that |ϕ(v ) − ϕ(v )| ∈/ T holds for all edges v v ∈ E. List If |L | = k for all i,thenL is termed a k-assignment. i j i j i T -colorings are defined in the natural way, choosing each The choice number of G (also called the list chromatic color ϕ(v ) from the corresponding list L .TheT -choice number in the literature), denoted χ (G), is the smallest i i number, i.e., the smallest k for which every k-assignment k such that every k-assignment L admits a list coloring. of G has a list T -coloring, will be denoted by χ (G). ≤ |T For χ (G) k, G is said to be k-choosable. Since the identical lists (defining L := {1,...,k} for all i)forma Note that a (list) T -coloring is required to be a proper i ∈ particular k-assignment, it follows by definition that the coloring if and only if 0 T ; in fact, a list coloring is a list T -coloring with T = {0},andχ = χ holds. |{0} chromatic number χ(G)ofG does not exceed χ (G). The concept of precoloring extension lies between k- colorability and k-choosability. In this problem, a vertex 0.4 Small uncolorable graphs subset W ⊂ V of the graph G =(V,E)isprecolored with We close this introduction with some simple examples ad-

ϕW : W →{1,...,k} mitting no list coloring, to illustrate the above definitions. for some k ∈ IN, where the mapping ϕW is not required Example 0.1. The complete bipartite graph K2,4 with to be onto (and, in particular, W = ∅ is also allowed), the lists {1, 2} and {3, 4} in the first vertex class and and the question is whether G admits a proper k-coloring {1, 3}, {1, 4}, {2, 3}, {2, 4} in the second class admits no

3 list coloring, hence it is not 2-choosable. Similarly, K3,3 without any gaps, take k new vertices u1,...,uk and join with the lists {1, 2}, {1, 3},and{2, 3} in each vertex class ui with vj if and only if i/∈ Lj (1 ≤ i ≤ k,1≤ j ≤ n). has no list coloring, therefore it is not 2-choosable either. Then, forgetting about the list assignment, precolor the On the other hand, it is easy to show that both graphs vertex ui with color i, for all i =1,...,k. This precoloring 2 4 3 3 are 3-choosable, thus χ (K , )=χ (K , )=3holds. of the larger graph is extendable with color bound k (i.e., without taking any new colors) if and only if G is list Example 0.2. One of the simplest non-3-choosable, pla- colorable. nar, K4-free graphs is obtained from K2 18 by inserting a , Precoloring extension vs. chromatic number. Let matching of nine edges in the 18-element vertex class. De- the graph G =(V,E) with precolored set W and color note these edges by e ,where1≤ i ≤ 3and4≤ j ≤ 6. ij bound k be given. Assuming that W ⊆ W is the (possibly Assign the lists {1, 2, 3} and {4, 5, 6} to the vertices in the i empty) set of vertices of color i for 1 ≤ i ≤ k, replace W 2-element class ; and the list {i, j, 7} to both vertices of i by a new vertex u (joining v ∈ V \ W to u if and each matching edge e . This 3-assignment admits no list i j i ij only if v had at least one neighbor in W ), and make the coloring. j i new vertices ui mutually adjacent, creating a complete Example 0.3. A non-3-choosable bipartite graph with subgraph of order k. The modified graph has chromatic number k if and only if the precoloring of G is extendable transparent structure is K7,7 , e.g. with the lists {1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 7}, {2, 5, 6}, {3, 4, 6}, {3, 5, 7} in with color bound k. each vertex class. These lists correspond to the seven List colorings vs. independence number. Given lines of the Fano plane, where the colors are viewed as the graph G =(V,E) with a list assignment L,construct points. It is well known (and easy to see) that if a set T the graph G2L with vertex set of at most three points meets all lines of the plane, then T itself is a line. Thus, in any selection of colors from the V (G2L):={(i, j) vi ∈ V,j ∈ Li} above lists, either at least four of the seven colors occur and join two of its vertices (i,j), (i,j) if and only if in each vertex class, or in one class the three colors of an they belong to the same vertex (i.e., 1 ≤ i = i ≤ n)or entire line are selected (and this line is a list in the other to the same color at adjacent vertices (j = j ∈ L ∩ L class, too). In either case, some color is selected in both i i and v v ∈ E). classes, implying that no list coloring exists because the i i corresponding two vertices are adjacent. Theorem 1.1. (Vizing [190]) The graph G =(V,E) with lists L admits a list coloring if and only if Example 0.4. Consider the list T -coloring problem on α(G2L)=n. K3,3 with lists {1, 2}, {1, 3}, {2, 3} in one vertex class and {1, 3}, {1, 4}, {3, 4} in the other class, where T = {2}. As a matter of fact, slightly more is true ; namely, there Though 0 ∈/ T , no feasible coloring exists. (Compared to is a bijection between the admissible list colorings and Example 0.1, the lists are now ‘ shifted ’ by 2 (mod 4).) the independent sets of cardinality n,asthevertexset The graph remains uncolorable even if we remove the two of G2L is partitioned into the n cliques induced by the { } { } { } { } edges ( 1, 3 , 1, 3 )and( 2, 3 , 1, 4 ). sets {(i, j) j ∈ Li},1≤ i ≤ n. Note further that if all lists are identical, then the above construction results in the known equivalent definition of the chromatic number, 1 General results stating that a graph G =(V,E)isk-colorable if and only if the ‘ Cartesian product ’ (also called ‘ box product ’) of In this section we review some of the most general facts, G and Kk has independence number |V |. walking around the subject from several different sides. 1.2 Complete bipartite graphs and the 1.1 Equivalent formulations construction of Haj´os Next, we present two types of reductions, taken from [18] Next, we consider complete bipartite — and more gener- and [190], respectively. The first one shows in two steps ally, complete multipartite — graphs, present estimates that the three problems of list coloring, precoloring exten- on their choice numbers, and show how they can be sion, and chromatic number are quickly reducible to each taken as building blocks to construct all non-k-choosable other. (In one direction it is obvious that, in general, graphs. list coloring is hardest and chromatic number is the most We have already seen (cf. Examples 0.1 and 0.3) that particular case, with all lists identical and having no pre- some bipartite graphs are not 2-choosable. As a matter colored vertices.) The second construction will establish of fact, the choice number of K tends to infinity with a relationship between list colorability and independence n,n n, and its growth can be described fairly accurately along number. the following observations of [62].

List colorings vs. precoloring extension. Assuming Denote by mr the minimum number of edges in an that a graph G =(V,E) with a list assignment L is given, r-uniform 3-chromatic hypergraph Hr (i.e., |H| = r for and that the union IL of the lists is the interval {1,...,k} all H ∈Hr, and in every vertex partition of Hr into two

4 parts, at least one part contains some H ∈Hr). View in each of its r classes. Alon [3] proves that there exist the vertices of Hr as colors, and assign the edges of Hr positive constants c1,c2 such that to the vertices in each vertex class of Kn,n, for any n ≥ 1 ≤ ∗ ≤ 2 mr, as lists. If there were a list coloring (in which no c r log t χ (Kr t) c r log t (1) color appears in both classes of K ), it would yield a n,n holds for every r and t. 2-partition of Hr with no part containing any H ∈Hr ; thus, χ (Kn,n) >r. On the other hand, if 2n

1 Consider the following three types of operations. 2 mr complete bipartite graph. p =4,q ≤ 18), F¨uredi, Shende and Tesman [169] (p =5, q ≤ 12), and O-Donnel [151] (p =6,q ≤ 10). In this way, the role of complete graphs is taken by the A related problem is to determine the smallest nk, complete bipartite ones when χ is replaced by χ .It k ∈ IN, for which there exists a non-k-choosable Kp,q with is interesting to note that, though there is an increas- p + q = nk. By the observations above, nk ≤ 2mk holds, ing number of (inclusionwise) minimal complete bipartite and for small values the bound is tight : n2 =6andn3 = non-k-choosable graphs as k gets large, all of them are 14 (the latter proved in [90], cf. Example 0.3). Hanson, equivalent from the generative point of view. MacGillivray and Toft also give the recursive estimate ≤ k nk knk−2 +2 which, for k even, provides a better 1.3 Typical behavior of the choice num- upper bound than the known ones for 2mk (e.g., n4 ≤ 40 is obtained, while the currently best upper bound on m4 ber is 23). Nevertheless, as sufficiently strong lower bounds In this subsection we present asymptotic results on the on mk are not available, the equality of nk and 2mk has choice number of random graphs and random bipartite not been ruled out for any value of k so far. graphs. For p much bigger than q, Hoffman and Johnson [98] On one hand, putting r := χ(G)andt := n,the determine the choice number, proving that inequality (1) implies

q−1 q−1 q χ (Kp,q)=q for (q − 1) − (q − 2) ≤ p

5 for p =1/2 is due to J. Kahn (its proof appeared in [4]) ; It is widely believed that the answer is affirmative (justi- the general case has been proved by Tuza and Voigt [182]. fying the word ‘ conjecture ’ in the title of this subsection), (The weaker upper bound of o(n), conjectured in [62], was and almost all known proofs showing that a certain graph first proved by Alon [3].) is (k, 1)-chosable can be extended with little effort to ver- ify (km, m)-choosability. Nevertheless, (k, )=(2, 1) is Theorem 1.4. For every fixed edge probability p, the only case for which the implication formulated in Problem 1.5 has been proved for all m and for all graphs G · χ (Gn,p)=(1+o(1)) χ(Gn,p) satisfying the supposition (i.e., for all 2-choosable graphs). This result, published in [186], can be extended to obtain with probability 1 − o(1) as n →∞. a reduction method as follows.

An important result of Bollob´as [25] states that Theorem 1.6. (Tuza, Voigt [187]) Let L be a k-    assignment on G =(V,E), and suppose that X ⊂ V 1 1 n is a vertex set such that the edges incident to X form χ(G )= + o(1) log , n,p 2 1 − p log n a 2-choosable graph. Then, there can be chosen a color ϕ(vi) ∈ Li for each vi ∈ X, in such a way that i.e., the expected value of the chromatic number asymp-    ∩{ ∈ ∈ } ≤ totically equals the order n divided by the expectation of Lj ϕ(vi) vi X, vivj E 1 the independence number. As regards the choice num- holds for every vj ∈ V \ X. ber, one can prove that there exists a slowly decreas- ing sequence n → 0 (the appropriate speed of conver- If a set X ⊂ V with the above property exists in G and, gence can be read out from numerical estimates of [25]) in addition, the induced subgraph G−X canbeprovento for which the following procedure successfully finds a list be (k − 1)-choosable, then the k-choosability of G follows coloring for any k-assignment with k =(1+n) · χ(Gn,p). as well. The (km, m)-choosability of G can be deduced in As long as there exists an independent set S of at least a similar way ; for instance, the (3m, m)-choosability of − · (1 n) α(Gn,p) currently uncolored vertices and a color the Petersen graph is obtained for every m ∈ IN . i appearing in the lists of all vertices in S, assign i to the entire S and remove i from all the other lists. On Graph union. One of the interesting consequences of the other hand, if such a large uncolored S does not exist an affirmative answer to Problem 1.5 (if it holds true in- anymore, then, for every subset Y of the currently un- deed) would be that the choice number is a submulti- colored vertices, the union of the modified lists belonging plicative function with respect to graph union. For the to Y contains at least |Y | colors, thus the remaining lists time being, however, this can only be formulated as yet have distinct representatives by the K˝onig–Hall theorem. another intriguing open problem.

It remains an open problem to settle whether χ(Gn,p) Conjecture 1.7. For any two graphs G1 and G2 on the and χ (Gn,p) have the same asymptotic behavior for every same vertex set, ‘ reasonable ’ edge probability function p = p(n). Neither χ (G1 ∪ G2) ≤ χ (G1) χ (G2) . (2) is it known how strongly χ is concentrated, and whether χ >χholds with probability 1 − o(1). To see that the implication of Problem 1.5 would indeed

Random bipartite graphs. Erd˝os, Rubin and Tay- imply (2), assume χ (Gi)=ki for i =1, 2. Starting with ∪ lor [62] investigated the random bipartite graph Bn,p with any (k1k2)-assignment of G1 G2,choosek2-element color m = n/2 vertices in each class and with edge probability sets Si ⊆ Li such that Si and Sj are disjoint whenever p =1/2 They proved the logarithmic growth of vivj ∈ E(G1) — on applying that the k1-choosability of G1 implies its (k1k2,k2)-choosability as well — and then log m 3logm find a list coloring of the k2-choosable graph G2 in the list <χ (Bn,p) < log 6 log 6 assignment (S1,...,Sn). (More generally, inserting the edges of a (b, c)-choosable graph into an (a, b)-choosable − →∞ with probability 1 o(1) as m . graph, we obtain an (a, c)-choosable one [62].) By the re- sults cited above, the inequality (2) holds (with equality) 1.4 Unions of graphs and the (am, bm)- if at least one of the two Gi is 2-choosable. conjecture Jensen and Toft [111] remark that so far (2) is not confirmed even for the following rather simple particular In this section we deal with some problems and results case. Suppose that G is bipartite, and substitute two related to (k, )-choosability. Perhaps the most challeng- nonadjacent vertices for each vertex of G. (Each edge of ing open question of this kind is the following one, being G becomes then an induced C4.) It is easily seen that unsolved for already almost two decades. the new graph G canbewrittenintheformG1 ∪ G2, where G1  G2  2G ; it is not known, however, whether ≤ 2 Problem 1.5. (Erd˝os, Rubin, Taylor [62]) If G is χ (G ) (χ (G)) . (k, )-choosable, does it follow that G is (km, m)- Let us mention here a further problem, that deals with choosable for every m ∈ IN ? the union of three graphs.

6 Conjecture 1.8. (Voigt [192]) Let G =(V,E)bea Problem 1.11. Characterize the structure of graphs graph with V = V1 ∪V2 ∪V3 where V1,V2,V3 are mutually G =(V,E) such that disjoint independent sets, and suppose that the subgraph ∪ ≤ ≤ | | induced by Vi Vj is 2-choosable for all 1 i

To see how small the sum χ (G)+χ (G) can be, consider Recently, Voigt proved in [194] that those graphs are 4- the complete r-partite graph G := Kr∗t of order√ n = rt, choosable, and more generally, (4m, m)-choosable for all with r := n/ log n vertex classes and t := n log n = ∈ m IN . n/r vertices in each class. Applying the upper√ bound Though the inequality (2) has not yet been proved, of Inequality (1), we obtain that χ (G)=O( n log n); an upper bound on the choice number of the union of two moreover, its complement G is just rK ,sothatχ (G)+ √ t graphs follows from a result of Alon (Theorem 2.5, to be χ (G)=O( n log n). (This construction in [3] answered discussed later). a question of [62] in the negative.) √ It is an open problem whether the factor log n is Theorem 1.9. (Alon) There exists a function h :IN× ≤ necessary√ in the formula, or perhaps χ (Gn)+χ (Gn) IN → IN s u c h t h a t χ (G1 ∪G2) ≤ h(χ (G1),χ (G2)) holds c n holds for an infinite sequence of graphs Gn of order n. for any two graphs G1,G2.

The superexponential upper bound read out from Theo- 2 Vertex degrees rem 2.5 is the best known general one, hence being very far from quadratic (as expected). In this section we discuss three main issues. The first one is to investigate the possible extensions of Brooks’s 1.5 Graphs and their complements theorem for various types of choosability, i.e., to obtain sufficient conditions in terms of vertex degrees for choos- The well known theorem of Nordhaus and Gaddum states ing colors or color sets from the lists. The second one is that a lower bound on χ (G) as a function of the average de- ≤ χ(G)+χ(G) n +1 gree, a property in which the choice number significantly holds for every graph G on n vertices. As shown in [62], differs from the chromatic number. The third and fourth this inequality can be strengthened to a far extent. subsections are devoted to an algebraic approach invented by Alon and Tarsi, that leads to sufficient conditions for Theorem 1.10. (Erd˝os, Rubin, Taylor [62]) Every choosability, in terms of the existence of certain orienta- graph G of order n satisfies tions on the edges. The bounds on edge colorings are also strongly related ≤ ≤ χ (G)+χ (G) col(G)+col(G) n +1. to vertex degrees, but we shall discuss them only later, in

− Section 3.3. For a short proof, denote by di the number of vertices v with jiwhich are nonadjacent to vi. Assuming that the vertices are labelled in a decreasing ≤ ≤ The inequality χ (G) col(G) Δ(G) + 1 yields an order of degree, we see obvious upper bound on the choice number. Certainly, − + the bound is tight, and one nice class attaining equal- d + d ≤ n − 1 ∀ i ≥ j. i j ity is that of the chordal graphs. In fact, arranging the − vertices of a chordal graph in reversed simplicial order (The inequality remains valid even if we replace di and + v1,...,vn (i.e., where for each i ≤ n, the neighbors vj d by the degrees dG(vi)andd (vj )intheentireG and j G of v with j

7 unless it is KΔ+1, or Δ = 2 and the graph is an odd cycle. see [35]. The corresponding result for hypergraphs ap- Erd˝os, Rubin and Taylor [62] and Borodin [30] strength- pears in [132]. ened this assertion, proving list colorability with lists of There are several results concerning ‘ critical lengths d(vi) for every vertex vi, provided that at least amenable graphs,’ too, where the lists are supposed to one 2-connected block of the connected graph is not a be nonidentical. See [39, 179] for further details and ref- clique or an odd cycle. Tuza and Voigt [185] showed fur- erences. ther that, under the same structural condition, color sets Sparse graphs. Perhaps the most involved theorems of cardinality m can be chosen whenever |Li| = md(vi) concerning vertex choosability vs. vertex degrees are re- for every vi. We summarize these results in the following assertion. lated to triangle-free graphs. The results summarized be- low are proved by a heavy use of probabilistic methods. Theorem 2.1. Let m ∈ IN, and let G =(V,E)bea The estimates nicely match with the general lower bounds connected graph. Suppose that L is a list assignment on the independence number, in terms of order and max- imum degree ([1, 166, 167, 168]). where |Li|≥md(vi)foreachvi ∈ V .If Theorem 2.3. Let G be any graph of maximum de- (i) |L | >md(v )forsomev ,or i i i gree Δ. (ii) G contains a block which is neither a complete graph (i) If G has girth at least 5, then χ (G) ≤ (1 + nor an induced odd cycle, Δ)Δ/ log Δ, where Δ → 0asΔ→∞(Kim [128]). ⊆ then G admits a choice of an m-element Ci Li for each (ii) If G is triangle-free, then χ (G) ≤ c Δ/ log Δ ∩ ∅ ∈ i, such that Ci Cj = for all vivj E. for some constant c independent of Δ (Johansson [113]). Further generalizations are known for list T -colorings (Waller [196], also making a distinction for the cases (iii) For every r ∈ IN there exists a constant cr such that ≤ where T is an arithmetic progression containing 0) and if G is Kr-free, then χ (G) (cr Δ log log Δ)/ log Δ colorings with respect to additive and hereditary graph (Johansson [114]). properties (Borowiecki et al. [35, 34]). The previous the- Apart from a multiplicative constant, the upper bounds orem does not hold true for infinite graphs, however, as in (i) and (ii) as functions of Δ are tight, since there exist shown by the following class of examples. Take the count- ∗ graphs of arbitrarily large girth with maximum degree Δ able star S with center v0 and leaves v1,v2,...,with the and chromatic number cΔ/ log Δ (see [24]). It remains list assignment L0 =INandL = {i} for all i ∈ IN , i an open problem to prove the asymptotic bound of (i) for and join v0 with a vertex of a finite 2-connected graph G the triangle-free case : which is neither a complete graph nor an odd cycle. If the lists on G are larger than |V (G)|, then the conditions Conjecture 2.4. (Kahn, Kim [128]) For triangle-free of the theorem are satisfied in the graph composed from graphs G of maximum degree Δ, S∗ and G, but no list coloring exists since already S∗ is ≤ uncolorable. χ (G) (1 + o(1)) Δ/ log Δ →∞ Critical graphs. A closely related classic theorem due as Δ . to Gallai [69] deals with the structure of subgraphs in- For relatively small maximum degree Δ ≥ 5andsuffi- duced by the set of vertices of minimum degree in a ciently large girth g with respect to Δ, the stronger ex- color-critical graph. To generalize this result, call a graph plicit upper bound χ ≤ Δ/2+2 was proved by Kostochka L G =(V,E) critical with respect to a color assignment [129, Remark 6]. It follows, in particular, that every graph if it has no list coloring, but each of its proper induced of maximum degree 5 and girth at least 35 is 4-choosable. subgraphs does have one. Clearly, |Li|≤d(vi)holdsfor every vertex vi if G is critical. Call vi small if its degree Almost disjoint lists. In the context of (p, q, r)- equals |Li|. choosability, upper bounds in terms of vertex degrees have been derived by Kratochv´ıl, Tuza and Voigt [138]. For in- Theorem 2.2. (Kostochka, Stiebitz, Wirth [132] ; Tho- stance, it is shown by probabilistic methods that if the massen [178]) If the graph G is critical with respect to lists are almost disjoint (say, r = p − c ) then lists of the list assignment L, then each block of its subgraph size 5.437 c Δ(G) always admit a list coloring and this induced by the small vertices is a complete graph or an bound is best possible for all c, apart from a multiplicative odd cycle. constant.

This result can be obtained directly from the proof of 2.2 Lower bounds on the choice number Erd˝os, Rubin and Taylor [62], too ; however, the new proofs are much simpler. In fact, Gallai’s original method The following result shows that χ is closely related to [69] can also be applied. Moreover, for general graph the essentially local parameter of vertex degree. In this properties P, the variations [36, 146] of Brooks’s and Gal- respect it essentially differs from the chromatic number lai’s theorems can be extended to list P-colorings as well, which is a global graph invariant in nature.

8 Theorem 2.5. (Alon [4]) Let k ∈ IN . I f infinity with d by Theorem 2.5, it follows that the choice      number of the planar unit distance graph is infinite. k4 k4 d>4 log 2 Making this assertion more precise, Schmerl [161] k k proved that the choice number for IR2 and IR3 is count- 2 1 able, as well as the ‘ rational distance graph ’ in IR ;and holds for the average vertex degree d := n (d(v1)+ ... that these bounds are not valid in higher dimension. +d(vn)) of G,thenχ (G) >k.

The proof is probabilistic, performed in two main steps. 2.3 Graph polynomials ⊆ Start with a bipartite subgraph H G of minimum de- The graph polynomial, also called the edge difference poly- ∪ | |≥| | gree at least d/4, with vertex partition A B, A B . nomial, of a graph G =(V,E) is defined as Simple calculation shows that selecting k-element lists  B k4 − for the vertices of from a -element color set IL ran-−1 PG = PG(x1,...,xn):= (xi xj ) k4 domly and independently, each with probability , i

9 thus P G ≡ 0andPG ∈I(Q1,...,Qn) ; and, conversely, Call an orientation G of G even if it has an even num- assuming PG ∈I(Q1,...,Qn), we obtain P G ≡ 0, thus ber of edges vivj with i>j, i.e., oriented from a vertex PG(x1,...,xn) = 0 for all (x1,...,xn) ∈ L1 ×···×Ln, of larger subscript to a smaller one ; and call G odd if and therefore G admits no list coloring. the number of those backwards-oriented edges is odd. To prove Theorem 2.9, one observes first that, writing PG as Uniquely list-colorable graphs. Dinitz and Mar- the sum of 2m (m := |E|) monomials, there is a bijection tin [49] analyze irreducible factors of the remainder P G between those 2m terms and the 2m possible orientations of PG modulo I(Q1,...,Qn), with emphasis on the case of G. (In the factor (xi − xj )ofPG,choosexi if the edge where G admits precisely one list coloring. For this pur- − vivj is oriented from vi to vj ,andchoose x j if it is ori- pose, it is convenient to view P G as a homogeneous poly- ented from v to v .) Hence, the monomials m xdi are | | ∪{ } j i i=1 i nomial of degree E over the set IL x1,...,xn of vari- in one-to-one correspondence with those orientations in ables. (Note that the substituting operation with respect which the out-degree sequence is (d1,...,dm). Thus, the to the Qi never destroys homogenity if also the colors are m di coefficient of =1 x in the standard representation of treated as variables.) It is proven in [49] that if (G, L) i i PG equals the difference between the numbers of even and admits precisely one coloring, say (c1,...,cn), then P G is odd orientations having out-degree sequence (d1,...,dm). the product of |E| linear factors,whereeachxi appears If two orientations G 1, G 2 have the same out-degree on the power |Li|−1, and the other |E| + |V |− |Li| sequence, then the set G 1⊕G 2 of edges oriented differently factors are of the form ci −cj.Whatismore,E admits an in G 1 and in G 2 is an Eulerian subgraph, and the parity orientation E for which a subset E ⊆ E can be chosen, of the number of its edges is even if both G 1 and G 2 are |  | n | |− E = i=1( Li 1), such that even or both are odd, and the parity is odd if precisely one     of G1 and G2 is even. Therefore, under the conditions of P G(x1,...,xn)= (xi − cj ) (ci − cj ) . m di Theorem 2.9, the coefficient of i=1 xi in PG is nonzero vivj ∈E vivj ∈E \E (as the mapping G → G ⊕ G is a parity-preserving bi- jection between orientations and Eulerian subgraphs if G | | Since P G has degree less than Li in each variable xi, is even, and parity-changing otherwise). Since all terms and all but one colors are infeasible at vi — setting PG to of PG have degree m and every reduction step (by which zero for each x = c, c ∈ L \{c }, the first product can i i i P G is derived from PG) decreases the degree of the mono- be equivalently written as m di mial to which it is applied, no new term i=1 xi can occur during the reduction steps ; and the original terms  n  m di | | (x − c )= (x − c) . i=1 xi in PG are not modified because di < Li is as- i j i ≤ ≤ ≡ =1 c∈Li sumed for all 1 i n.Consequently,P G 0, and thus vivj ∈E i c=ci G admits a list coloring. In particular, the formula establishes a bijection between Orientations without odd circuits. An interesting the edges of G and the irreducible factors of P G,foreach case, worth mentioning separately, is where eo(G )=0, uniquely colorable list assignment L on G. i.e., if no directed circuits of odd length occur in the ori- entation. Since ee(G ) > 0 (as the edgeless subgraph al- ways is Eulerian), Theorem 2.9 implies that the maximum 2.4 Orientations and Eulerian subdi- out-degree plus 1 is an upper bound on the choice num- graphs ber. For eo(G ) = 0, however, the algebraic machinery is not needed, as an elementary proof works by applying In general, it is not easy to check whether P can be G Richardson’s theorem [153]. This result guarantees that, expressed in terms of a combination of the Q with poly- i under the ‘ no odd circuits ’ assumption, G contains an nomial coefficients, therefore Theorem 2.8 is not a ‘ good independent set S such that from each v ∈ V \ S there is characterization’ in the algorithm-theoretic sense. One at least one edge oriented to some vertex of S.Insuch can deduce a more explicit sufficient condition for col- orientations, the method of the proof described for The- orability from it, however, with the help of orientations. orem 3.12 finds a list coloring whenever the out-degree of To formulate the result, call a digraph G Eulerian if the each vertex v is smaller than |L |. in-degree equals the out-degree for each of its vertices. i i In several situations, the following related observation (Hence, such a digraph is not required to be connected, turns out to be useful. and it is allowed to have an arbitrary number of isolated  vertices, too.) We denote by ee(G) the number of those Lemma 2.10. If G =(V,E)andd ∈ IN such that, for  spanning Eulerian subgraphs of G which have an even every t ≤|V |, each induced subgraph on t vertices has  number of edges, and by eo(G) the number of those with at most dt edges, then G has an orientation of maximum an odd number of edges. out-degree at most d.

Theorem 2.9. ([7]) Let a graph G =(V,E) with a col- This assertion seems to have been in the folklore at lection L of lists be given. If G has an orientation G such least from the second half of the 1980s ; a proof can be that the out-degree of each vertex vi is at most |Li|−1, found in [7]. By the observations above, if G is bipartite, and ee(G ) = eo(G ), then G is L-list colorable. then the lemma yields an orientation G with a guaranteed

10 upper bound not only on the maximum out-degree, but later) and some further results are summarized next. One ≤ also on the choice number. may note at the beginning that χ (G) 6 is easily seen, because every planar graph contains a vertex of degree at Corollary 2.11. Every 4-regular bipartite graph is 3- most 5, therefore col(G) ≤ 6alsoholds. chosable. More generally, for all k, m ∈ IN , e v e r y 2 k- regular bipartite graph is (km + m, m)-choosable. Theorem 3.1.

Contrary to the algebraic proof of Theorem 2.9, these (i) Every planar graph is 5-choosable (Thomassen ideas can be turned to a polynomial algorithm that finds [176]). a list coloring when the relevant assumptions hold. On the other hand, as noted by Jensen and Toft [111], there (ii) There exists a non-4-choosable planar graph seem to be no efficient algorithms known that find the (Voigt [191]). smallest possible maximum out-degrees in orientations G with ee(G ) = eo(G ) or in those with no odd directed (iii) Every planar graph is (4, 1, 3)-choosable (Kratoch- circuits. v´ıl, Tuza, Voigt [138]). 4-regular Hamiltonian graphs. One of the success- (iv) Every triangle-free planar graph is 4-choosable. ful applications concerns graphs with 3t vertices and 6t edges, whose edge set is the union of a Hamiltonian cy- (v) There exists a non-3-choosable triangle-free planar cle and t vertex-disjoint triangles. For such graphs, Du graph (Voigt [193]). and Hsu [52] conjectured that the independence number equals t, and Erd˝os raised the problem whether they al- (vi) Every triangle-free planar graph is (3, 1, 2)- ways are 3-colorable. This has been answered in the fol- choosable (Kratochv´ıl, Tuza, Voigt [138]). lowing stronger form. (vii) Every planar graph of girth 5 is 3-choosable (Tho- Theorem 2.12. (Fleischner, Stiebitz [67]) If a directed massen [177]). graph G is the edge-disjoint union of a Hamiltonian cir- (viii) Every bipartite planar graph is 3-choosable (Alon, cuit and some mutually vertex-disjoint, cyclically oriented Tarsi [7]). triangles, then ee(G) − eo(G ) ≡ 2 (mod 4), and, conse- quently, the underlying graph of G is 3-choosable. Further constructions for parts (ii) and (v) were found by Without applying the algebraic machinery of Theo- Gutner [82]. Moreover, as noted in [195], a construction rem 2.9, Sachs [157] presents a purely combinatorial proof of [82] (as well as one of [195]) is a non-4-choosable planar for the weaker assertion of 3-colorability. graph of chromatic number 3, having an uncolorable list assignment on as few as |IL | = 5 colors. The currently List T -colorings. Recently, Alon and Zaks [9] gener- known smallest 3-colorable non-4-choosable planar graph, alized Theorem 2.9 for list-T -colorings. They consider with 63 vertices, is presented by Mirzakhani [148] (also multigraphs Gm whereeachedgeofG is replaced by describing the interesting story of ‘ teamwork ’ how the 2|T |−1 parallel edges if 0 ∈ T ,andby2|T | parallel edges record of 63 has been achieved). In her construction, too, if 0 ∈/ T . Then, if Gm admits an orientation G m where an uncolorable list assignment with |IL | =5isgiven.  m   m ee(G ) = eo(G ) and the out-degree of each vertex vi is To (viii), one may note that K2,4 is bipartite, planar, smaller than Li,thenG admits a list-T -coloring. and not 2-choosable. Furthermore, the k-choosability re- sults (k =3, 4, 5) extend to (km, m)-choosability for all m ∈ IN. In connection with (iii), the following problem 3 Comparisons of coloring param- remains open. eters Problem 3.2. ([138]) Is every planar graph (4, 1, 2)- In the bulk of this section, we investigate graph classes choosable ? in which the choice number is not much larger than the chromatic number. Classical examples of this kind are Moreover, Skrekovskiˇ asks concerning (vi) whether there the planar graphs, while a fundamental open problem is exist any planar, non-(3, 1, 2)-choosable graphs. related to line graphs. At the end, we discuss the rela- The proofs of the various upper bounds on the choice tionship between subset choosability and the fractional number in Theorem 3.1 use quite different techniques. chromatic number. Part (iv), that belongs to the folklore and seems to have been first mentioned explicitly in [136], is just a simple 3.1 Planar graphs remark on applying Euler’s formula ; (vii) requires a lot of intermediate steps to verify ; (iii), (vi), and (viii) are Planar graphs have always been special objects in the based on the fact that the graphs in question admit an study of graph colorings. The paper by Erd˝os, Rubin and orientation with maximum out-degree 3 and 2, respec- Taylor [62], too, contained several challenging questions tively (cf. Lemma 2.10) ; and the proof of (i) is already a about them. The answers (each found more than a decade classic, that we present next.

11 The proof of 5-choosability. As the assertion is triv- 3.2 Graphs with equal chromatic and ial for graphs of order at most 5, one can apply induction choice number on n. We may assume that G is a 2-connected near- triangulation, i.e. all of its internal faces are triangles. Beside the asymptotic results of Section 1.3, it would be Omitting colors from lists on the outer cycle C,thefol- of great interest to know which conditions ensure that lowing induction hypothesis will be applied on the list the choice number equals the chromatic number. At the assignments : Two consecutive vertices v1,v2 of C are col- current state of the art, however, it seems hopeless to ored (i.e., |L1| = |L2| = 1) with distinct colors, lists at the find a characterization theorem for graphs G satisfying other vertices of C have size 3, and vertices not incident χ (G)=χ(G). to C have lists of 5 colors each. If C is a triangle, one Graphs of small chromatic number. Already the can immediately reduce G by omitting v1, v2,andtheir 1 case of 2-choosable bipartite graphs, settled by Rubin , colors from the lists of their neighbors (with just a little is not at all trivial. To formulate the result, define the more care if an internal vertex is adjacent to the entire core of G the subgraph obtained by successively removing C). Hence, we assume |C|≥4. vertices of degree 1 as long as such a vertex is present in ∈ If C has a chord, say vi,vj V (C) are adjacent but the current graph. Moreover, let us say for short that a { } nonconsecutive on C,then vi,vj splits G into two parts graph is a θ-graph if it consists of two vertices of degree 3 G1,G2,havingtheedgevivj on their outer cycles, and one joined by three paths of respective lengths 2, 2, 2m (m ∈ of them, say G1, contains the two colored vertices of C. IN arbitrary) all internal vertices of which have degree 2. Finding a list coloring of G1 by the induciton hypothesis, (I.e., one edge of K2,3 is subdivided into an odd path.) vi and vj get colored on the outer cycle of G2,andthen G2 is also list colorable. Theorem 3.3. (Rubin [62]) A connected graph is 2- If C has no chord, consider the uncolored neighbor of choosable if and only if its core is either a single vertex or v2,sayv3, and reduce its list to a 2-element subset L3 an even cycle or a θ-graph. not containing L2.SinceG is a near-triangulation and |C| > 3, the neighbors of v3 induce a path P from v2 to The smallest uncolorable 2-assignments of a non-2- the uncolored neighbor v4 of v3 on C,andP is internally choosable graph require at most four colors in IL. Hoff- disjoint from C (as C has no chord) ; therefore, the lists man, Johnson and Wantland [99] observe that under the of size 5 on P can be reduced to 3-element lists disjoint additional condition |IL |≤3, the graphs K2,n−2 (and only − from L3. Finding a list coloring of G v3 by induction, those) become 2-choosable, for all n. v4 is the unique vertex that can exclude one of the two It is worth noting here that the T -choice version of colors from L3, therefore G, too, is list colorable. list colorings seems to be much harder than the problem { } Defective colorings. Cowen, Cowen and Woodall [45] for T = 0 . Already for a subcase of k = 2, namely for consider vertex colorings ϕ which are not proper, but for cycles of even length, and for some rather restricted sets afixedd ∈ IN every vertex v has at most d neighbors T , unexpected difficulties arise. of color ϕ(v). In the list coloring version, call a graph ∗ Conjecture 3.4. (Alon, Zaks [9]) For every n, r ∈ IN , (k, d) -choosable if it admits such a coloring for every k- and for the set T = T := {0, 1,...,r}, assignment L. This concept was recently introduced in- r ˇ dependently and simultaneously by Skrekovski [170] and 4n − 2 Eaton and Hull [55]. In the manuscript [170], the follow- χ C2n = · (2r +2) +1. |T 4n − 1 ing collection of results is announced : In [9], the conjecture is proved for cycles of length four. (i) Every planar graph is (3, 2)∗-choosable. For a subclass of 3-colorable graphs, we mention the (ii) Every triangle-free planar graph is (3, 1)∗-choosable. following result.

(iii) Every outerplanar graph is (2, 2)∗-choosable. Theorem 3.5. (Gravier, Maffray [75]) Suppose that ω(G) ≤ 3 in the graph G =(V,E). If the edge set can (iv) Every triangle-free outerplanar graph is (2, 1)∗- be partitioned into two sets E ∪ E = E in such a way choosable. that each induced P3 of G has precisely one edge in each of E and E ,thenχ (G)=χ(G)=ω(G). The assertions (iii) and (iv) concerning outerplanar graphs have also been proved by Eaton and Hull. Both Further problems. Graphs with larger chromatic [170] and [55] ask whether every planar graph is (4, 1)∗- number are considered in recent works by Gravier and choosable ; if true, this would be an interesting general- Maffray. In [76] they investigate graphs in which there ization of a theorem of [45]. exists a k-coloring without color classes of more than 2 Answering a problem of [170] in the negative, Tuza vertices. Corollaries are derived for claw-free graphs (i.e., and Voigt have constructed 3-colorable planar graphs 1 ∗ There are several important results in the paper [62] attributed which are not (3, 1) -choosable. (One simple example is by its authors to A. L. Rubin who was working on his Thesis at that 27P3 +2K1.) time.

12 graphs containing no induced star of degree 3) of small or- 3.3 Edge and total colorings der, from which it follows that if G is the complement of There is a large number of results motivated by the List a triangle-free graph, then χ(G)=χ (G). An interesting related problem is Coloring Conjecture (Conjecture 3.10 below) which states the equality χ = χ for line graphs. In this subsection we survey the results related to this problem, but in the more Conjecture 3.6. (Gravier, Maffray [75, 76]) If G is convenient terminology of edge and total colorings, rather claw-free, then χ (G)=χ(G). than coloring line graphs and total graphs. In some sense, this conjecture seems to be ‘ too strong,’ We shall use the following notational conventions, and perhaps it would be worth making further efforts to analogously to vertex colorings. find a counterexample. On the other hand, if it turns out Prime notation. The parameters corresponding to to be true, then it implies the validity of the famous List chromatic and choice numbers for edge colorings are de- Coloring Conjecture, too. (The latter will be discussed in noted in the same way, except that we write χ instead of the next subsection.) χ, as follows :

Choice-perfect graphs. Motivated by the concept of χ(G)=thechromatic index of G, perfect graphs, one can define various types of ‘ choice χ (G)=theedge choice number or list chromatic index perfectness,’ and raise the following problem. of G = the smallest k such that every k-assignment Problem 3.7. ([181]) Characterize those graphs G in L on the edges of G admits a list coloring. which χ (G)=f(G) holds for every induced subgraph These parameters are just the corresponding values of G,where χ(L(G)) and χ (L(G)) of the line graph L(G)ofG. (i) f(G ):=χ(G ), Double prime notation. The parameters for total col- orings are denoted by χ with the analogous subscripts : (ii) f(G):=ω(G). χ(G)=thetotal chromatic number of G = the smallest The second property implies that G is perfect, but the number of colors in a proper coloring of V ∪ E, first one doesn’t ; for instance, the odd cycles are ‘ perfect ’ χ(G)=thetotal choice number (or the total list chro- in the sense of (i). Further choice-perfect classes will be mentioned in the next subsection. matic number) of G = the smallest k such that every k-assignment L on V ∪ E admits a list coloring. Concerining the equality χ = χ , the following prob- lem extends the famous Erd˝os–Lov´asz–Farber conjecture The following lemma, the variants of which have been (see e.g. [60, p. 26]) for choosability. observed by many authors, shows that total list colorings are closely related to the edge choice number. Conjecture 3.8. (Alon [5]) If G is the edge-disjoint union of n complete graphs of order n each, then Lemma 3.9. For every graph G, χ(G) ≤ χ (G)+2. χ (G)=n. The key idea of the proof is to color the vertices first. This can be done, for any k-assignment with k = χ (G)+2, by Kahn [124] has observed that a slight modification in the proof of the main result in [122] yields χ = n + o(n) the inequalities χ(G) ≤ Δ(G)+1 <χ (G)+2. Removing for these graphs. From another point of view, motivated the vertex colors from the list of each edge, at least χ (G) by Theorem 2.9, Alon and Seymour [5] have proved that colors remain in each list, so that a total list coloring such a graph always has an orientation with maximum exists. In this way, every upper bound on the edge choice out-degree at most n − 1. number yields one on the total choice number as well. As accounted in [86], the following problem has been A theorem on matroids. In the context of χ = χ,we raised independently by several authors, including Vizing, mention the following result of Seymour [164] who derives Gupta, Albertson and Collins, and Bollob´as and Harris. it from the Matroid Union Theorem [57, 150]. Let M be a matroid whose set X of elements can be partitioned into Conjecture 3.10. (List Coloring Conjecture) For | |≥ k independent sets. If Lx is a set with Lx k for each every multigraph G, χ (G)=χ(G). x ∈ X, then there exists a partition of X into independent ∈ ∈ sets Xi, i x∈X Lx, such that i Lx for all i and all A challenging related recent problem has been raised by ∈ x Xi. several authors independently (Borodin, Kostochka and It follows, in particular, that if the edge set of a graph Woodall [32] ; Juvan, Mohar and Skrekovskiˇ [119] ; Hilton can be decomposed into k forests, and each edge is as- and Johnson [95]). For general reference, we propose a signed to a list of k colors, then a color can be chosen name for it. for each edge from its list so that no cycle is monochro- matic. (Certainly, colorings obtained this way are usually Conjecture 3.11. (Total Choice Conjecture) For every multigraph G, χ(G)=χ(G). not proper edge colorings.)

13 Subdividing each edge of G into a path of lenght 2, we there exist edges oriented from each e ∈ Ei \ Mi to some 2 obtain a graph H whose square H is isomorphic to the e ∈ Mi. Assign color i to the members of Mi,remove 2 2 ‘ total graph ’ of G,sothatχ(H )=χ (G)andχ (H )= i from the lists of Ei \ Mi, and delete all edges oriented χ (G). In this direction, Kostochka and Woodall [133] from Ei \ Mi to Mi. Since a list gets shortened only if generalize the Total Choice Conjecture to the following the corresponding out-degree is decreased, all lists remain one, that we may call the ‘ Square Choice Conjecture ’ : longer than the out-degrees, and eventually the entire G 2 2 For every graph G, χ (G )=χ(G ). becomes edge-colored. For multigraphs, one needs an ex- Though the List Coloring Conjecture is still open in tension of the Stable Marriage Theorem, which follows general, considerable progress has been achieved. A triv- immediately by a more general result of Maffray [143]. ial upper bound is χ (G) ≤ col(L(G)) < 2Δ(G). After Earlier results and extensions. Galvin writes very the subsequent improvements by Bollob´as and Harris [26], modestly in his Introduction : “ The proof is very sim- Chetwynd and H¨aggkvist [41] (for triangle-free graphs), ple and uses no new ideas.” Nevertheless, his theorem and Bollob´as and Hind [27], Kahn [121, 122] proved the settles the long-standing conjecture of Dinitz (raised in asymptotic result 1978, also cited in [62]) which is just the rather particu- lar case G = K . Before Theorem 3.12, Janssen [110] χ (G)=Δ+o(Δ) n,n solved the problem for all unbalanced complete bipartite graphs, proving χ (K )=max(p, q) for all p = q. (She by the ‘ incremental random ’ method, not only for all p,q graphs of maximum degree Δ(G)=ΔasΔ→∞, but proved that, with a suitably chosen out-degree sequence also for families of hypergraphs of maximum degree Δ d, L(Kp,q) admits just one orientation without cyclic tri- where each pair of vertices is contained in a sufficiently angles, while the even and odd orientations — cf. the first small number (i.e., o(Δ) ) of (hyper)edges with respect paragraph after Theorem 2.9 — containing at least one to Δ. cyclic triangle can be matched with each other by a bijec- So far the estimate with best known error term for tion. Consequently, the monomial corresponding to d in the standard representation of P has coefficient 1 or −1, graphs seems to be G  implying list colorability. This idea was developed fur- 2/3 ther in [87] for the proof of the upper bound χ (K ) ≤ n χ (G)=Δ+O(Δ log Δ) , n cited above, to match even and odd orientations which proved by H¨aggkvist and Janssen [87]. They also prove, are not transitive on some clique in a fixed clique decom- by an involved application of Theorem 2.9, that position of a given graph.) Previous significant progress was achieved by H¨aggkvist [84], for the case p ≤ 2q/7. A χ (K ) ≤ n, n self-contained presentation of the proof of Theorem 3.12 can be found in [171], and further sufficient conditions which is in fact best possible for n odd. A more restricted for list edge colorability (where the conditions on the version of list total colorings of Kn, where the number edges are given by lists on the vertices, strongly moti- of occurrences of the colors is also prescribed, is due to vated by problems on Latin squares) have been published Sun [172] (proving a conjecture of [41]). by H¨aggkvist [85]. Line graphs of bipartite graphs. The following cele- It is immediately seen that the (m Δ(G),m)- brated theorem settles Conjecture 3.10 for all cases where choosability of the line graph of any bipartite multi- G is bipartite. graph G follows by the same argument for every m ∈ IN . Borodin, Kostochka and Woodall [32] extend this method Theorem 3.12. (Galvin [70]) If G is a bipartite multi- to prove that if each edge e = xy of G hasalistofatleast graph, then χ (G)=χ(G)=Δ(G). max (d(x),d(y)) colors, then L(G) admits a list coloring. Note further that Galvin’s theorem implies χ(G) ≤ If G has no multiple edges, then the surprisingly simple Δ(G) + 2 for every bipartite multigraph G. It is conjec- argument just combines the ‘ Stable Marriage Theorem ’ tured in [32] that a total list coloring exists already for of Gale and Shapley [68] and a useful idea of Bondy, Bop- edge-lists of length Δ(G) + 1, provided that all vertices pana and Siegel [28], as follows. Start with a proper edge have lists of Δ(G) + 2 colors. The converse (when only coloring ϕ : E →{1,...,Δ(G)}. Denoting by X and Y the vertex-lists are shortened to Δ(G) + 1) always admits the two vertex classes of G, for each incident edge pair e, e a list coloring, as shown above. with ϕ(e) >ϕ(e), orient the edge ee ∈ E(L(G)) from e to e if e∩e ∈ X,andfrome to e if e∩e ∈ Y .Inthisori- Nonbipartite multigraphs. Multiple edges seem to entation, the maximum out-degree is at most Δ(G) − 1. create lots of extra difficulties. Until quite recently, the Assuming that the out-degree of each e is smaller than only improvement on the trivial upper bound of 2Δ was Hind’s unpublished inequality χ ≤ 9Δ/5in[97]. the number of colors in the list of e (which is certainly the case at the beginning in any Δ(G)-assignment), the Theorem 3.13. (Borodin, Kostochka, Woodall [32]) following procedure successfully list-edge-colors G :Tak- Let G =(V,E) be a multigraph, and suppose that the ing the colors i ∈ IL one by one, consider the set E ⊆ E i list of each edge e = xy ∈ E contains at least of those uncolored edges whose lists contain i.By[68],Ei   1 contains a matching Mi which is ‘ absorbant ’ in Ei, i.e., max (d(x),d(y)) + 2 min (d(x),d(y))

14 colors. Then Gadmits a list coloring. In particular, (iii) G is planar, χ (G) ≤ 3 Δ(G) . 2 then χ (G)=d. This result immediately implies Shannon’s tight bound The third part states that a d-regular planar multigraph [165] on the chromatic index of multigraphs  of given max- 3 has χ (G)=d if and only if χ (G)=d. For the case of d = imum degree. Moreover, χ (G) ≤ Δ(G) +2alsofol- 2 3, this yields that the Four Color Theorem is equivalent lows. However, this bound may not be tight : also to the assertion that every planar 2-connected cubic graph is 3-edge-choosable. As noted in [58], this follows Conjecture 3.14. ([32]) If G is a multigraph  of max- 3 already from the results of Scheim [160], that can in turn imum degree Δ > 4, then χ (G) ≤ 2 Δ .Moreover, be deduced by combining a theorem of Vigneron [188] (cf. if G is connected, not complete and not an odd cycle, 3 also [107]) with some ideas of Alon and Tarsi [7]. then every list assignment with edge-lists of size 2 Δ and vertex-lists of size Δ is colorable. Taking another view on graphs embedded in the plane, projective plane, torus, and the Klein bottle, List coloring analogues of several further questions can Borodin, Kostochka and Woodall [32] provide sufficient be raised, for instance whether χ ≤ Δ+μ +1whereμ conditions for the equalities χ (G)=Δ(G)andχ(G)= denotes the maximum edge multiplicity (conjectured in Δ(G) + 1 in terms of combinations of girth and maxi- [119]), or whether χ (G) does not exceed the largest of mum degree, extending the earlier results and methods of Δ(G)and Borodin [31]. The larger girth, the smaller vertex degree suffices. We recall here the case with unrestricted girth. { 2|E(H)| ⊆ | | } Δ (G):=max |V (H)|−1 H G, V (H) odd Theorem 3.17. ([32]) If a graph G of maximum de- (cf. [122, p. 12]). Explanation for the latter formula is gree Δ(G) ≥ 12 is embeddable in a surface of nonneg- that for multigraphs G, the fractional chromatic index ative characteristic, then χ (G)=Δ(G)andχ(G)= χ∗(G) equals max {Δ(G), Δ(G)}, by the Matching Poly- Δ(G)+1. tope Theorem of Edmonds [56] (cf. e.g. [162]). This bound The equalities χ = χ and χ = χ for outerplanar is asymptotically valid : graphs have been proved by Juvan and Mohar [117]. Theorem 3.15. (Kahn [123]) For the class of multi- The upper bound of Δ+1. Most of the results above graphs G, verify the List Coloring Conjecture for some graphs with χ (G)=(1+o(1)) max {Δ(G), Δ(G)} χ = Δ. Concerning the other case, χ =Δ+1,Juvan, Mohar and Skrekovskiˇ study the problem for small maxi- as Δ →∞. mum degree. They note that the upper bound χ (G) ≤ 4 for (simple) graphs with Δ(G) ≤ 3 is implied by the choice An attractive conjecture of Kahn [122, 123] states that version of the Brooks theorem (indeed, to create K4 in a the asymptotic equality of the edge choice number and line graph would require a vertex of degree at least 4 or the fractional chromatic index remains valid for r-uniform a triangle with a multiple edge), and prove in [118] the hypergraphs (or hypergraphs with maximum edge size r, stronger assertion that if a subgraph E ⊂ E of maximum ∈ possibly with multiple edges) as well, for every fixed r degree 2 has lists of size 3 and the edges of E \ E have IN, as, say, χ gets large. lists of size 4, then G is list colorable. Subsequently, they Some upper bounds on χ and χ in terms of Δ plus prove in [118] that every graph of maximum degree 4 is the maximum local average degree are presented in [32]. 5-edge-choosable. Their method is strongly based on the treatment of so-called ‘ half-edges ’ (those incident to just Regular graphs of class 1. Developing the algebraic one vertex), to which shorter lists are assigned, and so an method (cf. Sections 2.4 and 2.3) further, Ellingham and inductive proof becomes possible by cutting off a suitably Goddyn [58] analyze the combinatorial meaning of the chosen small subgraph. coefficients of the monomials in the expansion of the graph polynomial. Some of their results are summarized in the For unrestricted maximum degree, Kostochka [130] next theorem. In its second part, ‘ Kempe recoloring ’ proved that if G contains no cycle shorter than 8Δ(log Δ+ 1.1), then χ (G) ≤ Δ+1. means that in a proper edge coloring we interchange the two colors on a 2-colored cycle, and repeat this operation an arbitrary number of times. 3.4 Choice ratio and fractional chromatic number Theorem 3.16. (Ellingham, Goddyn [58]) Let G be a d-regular multigraph with χ(G)=d.If Motivated by Problem 1.5, the study of the set { k } (i) G has an odd number of edge colorings with d colors, CH(G):= G is (k, )-choosable or leads to some interesting observations. It was first proved (ii) any two of its edge d-colorings have a Kempe recol- in Gutner’s Thesis [81] (cf. also [4]) that the elements of oring to each other, or CH(G) can be arbitrarily close to χ(G).

15 The concept of fractional chromatic number admits a of P (proceeding clockwise), delete it from the 2nd, 4th, further strengthening in this assertion. Denote by S the ... lists, and also delete the edges incident to a vi when collection of all independent sets in G, and consider already t colors have been selected for vi. Repeating this  procedure for each color and each possible P sequentially, ∗ ∗ χ (G):=inf ϕ (S) , a subset of t colors will eventually be selected for every ϕ∗ ∈S S vi because only those (at most) t colors get deleted from the shortened list of size 2t which have been selected for where the infimum is taken over all functions vi−1. ≥0 ϕ∗ : S→IR As regards bipartite graphs, Tuza and Voigt [185] showed satisfying the condition that K2,4 is (2m, m)-choosable if and only if m is even,  and more generally they proved that the same property ϕ∗(S) ≥ 1 holds for every minimally non-2-choosable bipartite graph S∈S (unpublished, 1995). vi∈S ∈ for every vertex vi V . One can show that the infimum 3.5 The chromatic polynomial is in fact attained as minimum, and χ∗(G) — termed the fractional chromatic number of G — is a rational number. Given a graph G =(V,E)andalistassignmentL = (L1,...,Ln), denote by f(G, L)thenumberofL-colorings Theorem 3.18. (Alon, Tuza, Voigt [8]) For every ϕ: V → IL. Kostochka and Sidorenko [131] proposed the graph G, problem of studying the function

inf {r ∈ CH(G)} =min{r ∈ CH(G)} = χ∗(G) . F (G, k):= min f(G, L) , |L1|=...=|Ln|=k

Choosing -element color sets Ci ⊆ Li from a k- i.e., the minimum number of L-colorings taken over all ∗ assignment L of G, and defining ϕ (S(j)) := 1/ for each k-assignments L. (The maximum would obviously be kn, j ∈ IL , w h e r e S(j):={vi j ∈ Ci}, a fractional coloring for all k ∈ IN . ) ∗ of G with value k/ is obtained, proving that χ (G)isa Denoting by P (G, k)thechromatic polynomial of G, lower bound. The other direction for the infimum is not it is clear by definition that F (G, k) ≤ P (G, k)holdsfor hard to prove by probabilistic methods ; and for the min- every G and every k,andthenon-k-choosable k-chromatic imum it can be deduced from a theorem of Huckemann, graphs show that in some cases this inequality is strict. Jurkat and Shapley (mentioned in [73] and proved also in [6]) by showing that for every fixed t and r,iftheedge Theorem 3.20. (Donner [50]) For every graph G there size of a uniform hypergraph with t edges is divisible by a exists an integer k0 = k0(G) such that suitably chosen integer, then the hypergraph admits a ver- tex partition of ‘ zero discrepancy ’ (i.e., equi-partitioning F (G, k)=P (G, k) each edge) into r classes. This argument also yields that holds for all integers k ≥ k0. the minimum is attained for infinitely many pairs (k, ). We note further that the result remains valid in a very The starting point of the proof is an observation that al- general setting, for induced hereditary properties [147]. lows us to compute f(G, L) recursively for every L (not Theorem 3.18 yields that the implication given in only for k-assignments). For any e = vivj ∈ E,denoteby Problem 1.5 is valid for infinitely many m, for every fixed G/e the graph obtained by contracting e (i.e., replacing vi pair (k, ) with k/ ∈ CH(G). Moreover, consequences and vj by a new vertex v and joining v to each vertex ad- for the 3-chromatic graph described in Conjecture 1.8 fol- jacent to at least one of vi and vj )andG−e := (V,E\{e}). low, too. For G/e, define the list assignment L/e to be identical to The sufficient value obtained from hypergraph theo- L on V \{vi,vj },andLv := Li ∩ Lj for the contracted retic methods for the smallest pair (k, ) attaining χ∗(G) vertex. One can see that that is rather large ; the next example shows that it can be the f(G, L)=f(G − e, L) − f(G/e, L/e) smallest one expected. holds for all G, L,ande ∈ E.Toprovef(G, L) ≥ P (G, k) ∗ Example 3.19. The equality χ (C2t+1) = 2+1/t is easy for every k-assignment L, Donner considers a computa- to see. The following short argument shows that C2t+1 is tion tree based on the above recursion, and makes esti- (2t +1,t)-choosable for every t ∈ IN. Assuming that the mates on the values at its leaves (each leaf is an edgeless vertices v1,...,v2t+1 are labelled consecutively along the graph). The partial sums of those values are analyzed by cycle, suppose that {1, 2,...,2t +1}⊆IL in the (2t +1)- distinguishing between the leaves according to the num- L assignment , and that each color j>2t + 1 is missing ber of contractions on the computation tree from the root ∈ from at least one list. Remove the color i from Li if i Li, to the leaf in question. and remove an arbitrary color otherwise. The consecutive occurrences of any one color induce subpaths P in the Problem 3.21. For which graphs G is the function cycle. Select this color for the 1st, 3rd, 5th, ... vertices F (G, k) identical to the chromatic polynomial P (G, k)?

16 Kostochka and Sidorenko [131] have observed that this We shall proceed in the order of increasing difficulty, equality holds for all chordal graphs ; on the other hand, considering PrExt first, also presenting the known trans- it obviously does not hold for any G with χ (G) >χ(G). parent necessary and sufficient conditions ; the complex- In the latter case, it follows by Donner’s theorem that ity of k-LC and the results related to k-CH will be dis- F (G, k) is not a polynomial. (Since F (G, k)andP (G, k) cussed in the third and fourth subsections. Finally, we coincide on all sufficiently large values of k,theformeris shall discuss results on graph coloring games. a polynomial if and only if it is identical to the latter.) Before the results on restricted graph classes, we quote a theorem on the running time of general list col- oring algorithms. 4 Algorithmic complexity General upper bounds. The chromatic number of a In this section we discuss some algorithmic results. For graph is a hard-to-estimate parameter, and all known al- terminology not introduced here concerning algorithmic gorithms determining it exactly run in exponential time complexity, we refer to [71] or the more recent book [37]. with respect to the number n of vertices (even when Note first that, since Chromatic Number is a par- the graph in question is supposed to be 3-colorable). In ticular case of List Coloring (aswellasofPrecolor- particular, Lawler [140] proposes an inductive algorithm ing Extension), in general the NP-completeness of the that computes the chromatic number of G and of all its induced subgraphs,√ where the total number of steps is latter follows from that of χ immediately. On the other 3 n hand, though the reductions presented at the beginning bounded above by ( 3+1) times a polynomial of n.The of Section 1 imply that these problems are equally hard method is based on the theorem of Moon and Moser [149] who proved that no graph of order n can have more than as long as the class of all graphs is considered, this is not n/3 necessarily the case anymore for many nicely structured 3 independent sets maximal under inclusion. (One also subclasses. needs the fact that the maximal independent sets can be listed efficiently, see [180, 115].) Variants of this result, For convenience, let us formulate the algorithmic e.g. those in [104] and [65], enable us to improve on the questions as decision problems. Keeping previous nota- guaranteed running time of coloring algorithms when re- tion, the vertex set will be assumed to be V = {v1,...,v } n stricted classes of graphs are considered. What is more, throughout. We shall first consider Lawler’s method can be extended for list colorings as well, and the following result is valid. Precoloring Extension ( PrExt ) : Theorem 4.1. (Hujter, Tuza [106]) There exists a Instance : Graph G =(V,E), subset W ⊆ V of precol- polynomial p(x) and an algorithm A such that, for ev- ored vertices, precoloring ϕW : W → IN , color bound k. ery graph G =(V,E) and every list assignment L, Question : Does there exist a proper coloring ϕ with at √ A | | ·| |· 3 most k colors such that ϕ(v)=ϕW (v) for all v ∈ W ? (i) the algorithm decides in at most p( V ) IL ( 3+ 1)|V | steps whether or not G is list colorable ; √ 3 | | For lists of equal size, the problem is (ii) if G√is triangle-free, then ( 3+1) V can be replaced by ( 2+1)|V | in the upper bound ; k-List Coloring ( k-LC ) : (iii) and, for every fixed t ∈√IN a n d ε>0, there is an L 3 |V | Instance : Graph G =(V,E), list assignment = n0 = n0(t, ε) such that ( 3+1) can be replaced | | ≤ ≤ |V | (L1,...,Ln), with Li = k for all 1 i n. by (1 + ε) for every graph of order |V |≥n0 that Question : Does L admit a list coloring on G ? contains no induced matching of t edges.

The above bounds are similar to those for the chromatic The general case, where no restriction is put on the number, the only difference is the (necessary) presence of lengths of the lists, will be called List Coloring, abbre- the factor |IL |. viated LC. k-Choosability ( k-CH ) : 4.1 Precoloring extension

Instance : Graph G =(V,E). Below we summarize the known results, grouped accord- ing to graph classes. To make more sensitive distinc- Question : Does G have a list coloring for every k- tions, in some cases we shall impose restrictions on the L assignment ? precolored set W , too. For convenience, we shall assume that the monochromatic subsets of W are W1,W2,...,Wk Obviously, the first two problems belong to the class (some of them may be empty), and that they are labelled NP. On the other hand, it will turn out that k-CH is in a decreasing order of cardinality, |W1|≥... ≥|Wk|. located higher in the hierarchy of complexity classes. (A ThecaseofW1 = ∅ leads to the complexity of Chro- well known fundamental open problem is whether or not matic Number, the literature of which will not be sur- those types of complexity are indeed distinct.) veyed here ; i.e., we assume |W1|≥1 throughout. Unless

17 otherwise stated, the given time complexity refers to the Distance constraints on W . Thomassen [178] proved original PrExt problem ; ‘ linear ’ means O(|V | + |E|). for planar graphs G that if k ≥ 5 and the vertices of W The graph is said to be F -free if it contains no induced are sufficiently far apart (with respect to |W |), then every subgraph isomorphic to F . k-coloring of W can be extended to that of the entire G. This result has recently been strengthened considerably Bipartite graphs : NP-complete in general [103], also by Albertson [2], proving that a percoloring is extendable for |W | = 3 [22], on planar bipartite graphs with in either of the following cases : k =3andonP14-free bipartite graphs with k =5 [134], P6-free bipartite graphs with unbounded k (i) k>χ(G) and the distance between any two precol- [105] ; linear if k = 2 (trivial), on P5-free bipartite ored vertices is at least 4 ; graphs for any k [105], and on trees and forests [105, 109]. (ii) k>χ (G) and the distance between any two pre- Line graphs : NP-complete on line graphs of complete colored vertices is at least 3. bipartite graphs [42] ; polynomial on line graphs of multiforests [145]. In particular, in a planar graph, distance 4 and 3 suf- fices for the extendability of a partial 5-coloring and 6- coloring, respectively. Albertson proves analogous results Split graphs and complements of bipartite graphs : for the more general case, too, where W induces the union polynomial, of the same complexity as Bipartite of vertex-disjoint cliques of sufficiently large mutual dis- Matching, apart from a multiplicative constant tances. 2 5 [105] (fastest known algorithms of O(n . ), see One of the interesting questions raised in [2] is e.g. [101]). whether or not distance constraints have similar conse- 3 quences for list colorings. That is, if W is precolored, Interval graphs : O(n )if|W1| =1,andNP-complete lists of given length k>χ are associated to the precol- if just |W1| = 2 is assumed [19]. orless vertices, and we wish to extend the precoloring of P4-free graphs (cographs) : linear [17, 105, 109]. W to a coloring of the entire graph by choosing a color from each list, how large should then be the distances Permutation graphs : NP-complete, already for between the vertices of W ? In particular, what is the |W1| = 1 [108]. smallest distance (if any) that suffices for planar graphs and lists of length 5 ? Complements of Meyniel graphs : polynomial if | | W1 = 1 [105], by the results of Hertz [93], apply- Undecidable problems. Here we mention some re- ing the algorithms of Gr¨otschel, Lov´asz and Schri- sults on infinite graphs. Similarly to the finite case, one jver [78, 79]. (A graph is said to be a Meyniel graph can ask whether a given precoloring on a finite subgraph ≥ if each of its odd cycles of length 5 contains at is extendable to a proper k-coloring of the entire graph, least 2 chords.) with fixed color bound k.

Perfect graphs : polynomial if W3 = ∅ and |W2|≤1, Burr [40] investigates this problem for a class of and NP-complete otherwise [135]. graphs of fairly transparent structure, called doubly- periodic graphs. The vertices of such a graph G are ∈ ZZ ∈{ } The NP-completeness for |W3|≥1or|W2|≥2on labelled vij (i, j ,  1,...,n ), the subgraphs { } perfect (more explicitly, on bipartite) graphs follows im- induced by vij1,vij2,...,vijn — called cells — are iso- mediately from the results of [22, 134] for k =3.On morphic for all pairs i, j, any other edge joins neighboring the other hand, as mentioned in [105], the complexity cells (i.e., cells whose i and j differbyatmostone),and of PrExt is not known for several graph classes whose both mappings i → i+1 and j → j+1 are automorphisms structure is well understood, e.g. for unit interval graphs ; of G. neither PrExt with the additional condition |W1| =1 It is proved in [40] that, for every color bound k ≥ 3, for chordal (and, in particular, strongly chordal) graphs. there exists a doubly-periodic planar graph G of maxi- Here is another innocent-looking related problem : mum degree 4 and a finite precolored set such that it is undecidable whether the precoloring can be extended to Conjecture 4.2. (Woeginger [197]) On planar bipar- a k-coloring of G. Dukes, Emerson and MacGillivray [53] tite graphs, PrExt with k =3and|W1| = |W2| = generalize this result to homomorphisms G → H (Burr’s |W3| = 1 is solvable in polynomial time. theorem deals with H = Kk). They prove undecidability for every finite, non-bipartite H, and for several finite bi- Woeginger notes that the condition |W1| =1makesthe partite graphs H, too ; e.g., for H containing a cycle C problem straightforward to solve on this restricted class of length at least 6, such that there is a homomorphism for any other color bound. The polynomial instances will h: H → C with h(v)=v for all vertices v of C.Itremains be discussed further in the next subsection, where struc- open, however, to characterize which H make the prob- tural characterizations will be given for the extendability lem undecidable (and, in particular, to prove or disprove of precolorings. decidability if H is a tree).

18 4.2 Good characterizations For each G ∈Gand for each (proper) partial k-coloring of G, contract each precolored color class to one new vertex, There are some transparent conditions that can be and make those new vertices mutually adjacent. The class checked efficiently on fairly large graph classes and pro- of graphs obtained in this way from G will be denoted by vide good characterizations for the polynomial instances G∗. It has been observed in [105] that if every G ∈Gis listed above. Most of them are collected in the paper by perfect, and for every precoloring of every G ∈Gthe core Hujter and Tuza [105] ; and an efficient general method condition is sufficient for precoloring extendability, then for perfect graphs with restricted precolorings has been every G∗ ∈G∗ is perfect, too. Perfect graphs satisfying developed by Kratochv´ıl and Seb˝o [135]. this requirement are called PrExt-perfect in [105]. Their Core Condition. A nonempty set U of pairwise ad- characterization — as well as that of the corresponding jacent precolorless vertices is called a q-core if there are class obtained by contraction — remains an open prob- lem. at least q −|U| distinct monochromatic classes Wi ⊆ W such that each vertex u ∈ U has at least one neighbor One of the interesting cases is the class G of P4-free in each of those Wi.If|U| =1,thenU is also called an graphs (cographs). Recently, Van Bang Le [141] described ∗ elementary q-core. The Core Condition requires that the G for them. It follows, in particular, that the member- precoloring of G contains no (k +1)-core. ship in this class can be decided in polynomial time. (The cographs themselves can be recognized in linear time, see Sequence Condition. Starting with a partial k- [43].) A characterization in terms of forbidden subgraphs, coloring of G, repeat the following procedure until it ter- however, is not known so far. minates. If there is a (k+1)-core or there exists no elemen- tary k-core, then stop ; otherwise choose an elementary Good characterization for PrExt on perfect k-core {u}, and assign to it the unique color not appear- graphs. We close this subsection with the strongest ing in its neighborhood. The Sequence Condition requires known related result on the general class of perfect graphs. that such a procedure must not result in a (k +1)-core. For a vertex v ∈ V and a collection H of not necessarily distinct subsets of V , dH(v) denotes the number of those Independence Condition. For each precolored class sets in H which contain v.Theterm‘ω-clique ’ means Wi and each precolorless vertex set U,denotebyα(U, i) ‘ complete subgraph on ω(G) vertices.’ the largest number of those mutually nonadjacent vertices in U which have no neighbor in Wi.TheIndependence Theorem 4.4. (Kratochv´ıl, Seb˝o [135]) Let G = | |≤ k ⊆ \ Condition requires U i=1 α(U, i) for all U V W . (V,E) be a perfect graph and X, Y ⊆ V two disjoint independent sets. Then G has a proper coloring ϕ: V → It is easily seen that each of the above conditions is {1,...,ω(G)} with the properties that X is monochro- necessary for the extendability of a precoloring if the color matic and ϕ(y) = ϕ(X) for all y ∈ Y , if and only if bound is k. The next statement summarizes the known cases where they are sufficient as well. |Q| ≥ |K| + |X|

Theorem 4.3. For the extendability of any partial col- holds for every multi-family Q of cliques and every family oring with color bound k in an instance of PrExt, K of at most |V | distinct ω-cliques satisfying ∀ ∈ \ ∪ (i) The Core Condition is necessary and sufficient for dQ(v)=dK(v) v V (X Y ) split graphs, complements of bipartite graphs, P4- and free graphs, and, if no color is repeated in W ,then dQ(v)=dK(v)+1 ∀ v ∈ X. also for complements of Meyniel graphs.

(ii) The Sequence Condition is necessary and sufficient The polynomial-time algorithm finding a required color- for forests, and, if k = 2, then also for bipartite ing when it exists is combinatorial, except for the only graphs. part that it calls for a maximum clique (for which no com- binatorial algorithm of polynomial running time is known (iii) The Independence Condition is necessary and suffi- so far on perfect graphs). For the particular case of Y = ∅, cient for line graphs of multiforests. this result answers a problem of Seymour who proved that it is NP-complete to decide whether two independent sets Part (iii) has been re-stated from the paper by Marcotte X, Y of unrestricted cardinalities in a perfect graph admit and Seymour [145], the other results appeared in [105]. a proper coloring with ω(G) colors such that X and Y are The case of interval graphs, with the assumption that no contained in distinct color classes [163]. color is repeated in the precoloring, admits a character- ization in terms of a Menger-type condition on directed graphs (constructed from the corresponding instance of 4.3 List colorings PrExt) ; see [17, 105] for details. On dense graphs, even with a very transparent structure, PrExt-perfect graphs. Motivated by the Core Con- the List Coloring problem is quite hard. In fact, as dition, the following graph operation can be introduced. Jansen and Scheffler [109] prove, it is NP-complete al- Let G be a graph class closed under induced subgraphs. ready on complete bipartite graphs, despite it is solvable

19 in linear time on every graph without induced subgraphs color assignments of X which can be extended to a list P4 if the total number |IL | of colors is bounded. Also, coloring of GΦ. (Choose color x for a variable vertex x if Kubale [139] observes that the NP-completeness of LC and only if the variable x is false in the truth assignment ; on line graphs of complete graphs follows from that of and, conversely, let x be false in Φ if and only if the color the Chromatic Index problem [100]. (In [139], LC is x has been chosen for x in a list coloring of GΦ.) Hence, shown to be NP-complete for bipartite graphs, too, un- each list coloring uniquely determines a truth assignment, der the further restriction that |IL | = 5 holds.) Recently, but not vice versa, because in some truth assignments Jansen [108] proved NP-completeness for the union of two some clauses are satisfied by more than one variable, each complete graphs. It is a natural related question to inves- allowing a distinct color choice. tigate which are the sparsest hard instances for LC. By this construction, the various theorems on Satis- fiability (e.g., on 3-SAT) yield NP-completeness results Polynomially solvable cases. In both early papers on list colorings restricted to the corresponding graph [190, 62] it is observed that 2-LC is easy to solve. Indeed, classes. Note further that edges may be added to GΦ one can obtain a linear-time algorithm by simply guess- in an arbitrary way as long as it remains bipartite, and ing the color ϕ(v)ofavertexv and check what sort of still the two-way mapping between colorings and truth as- implications this color would have for the other vertices. signments is preserved. It follows, for instance, that LC If ϕ(v) occurs in the list of some neighbor u of v,thenu is NP-complete on 3-regular bipartite graphs. gets forced to be asssigned to the other color of its list ; and this forcing step may be repeated for the neighbors Note that also the degree condition in Theorem 4.5 of u, etc. If this procedure stops when a subgraph G is quite strong when compared to the chromatic number is properly colored while all uncolored vertices still have problem. In fact, by applying the theorem of Brooks, we two colors in their lists, then G is list colorable if and obtain that χ(G) = 3 can be decided in linear time for ≤ only if so is G − G. On the other hand, if ϕ(v)leadstoa graphs of maximum degree 3, since χ(G) 3 holds if and contradiction (excluding both colors from the list of some only if G contains no connected component isomorphic to vertex), then in any list coloring of G (if it exists), the K4. only choice for v can be the other color, which then either Colors in a bounded number of lists. For longer leads to a final contradiction or reduces the problem to a lists, the easy and hard instances can be separated in smaller subgraph in linear time. terms of bounds on the number of how many times a color Further easy instances include the graphs of maxi- may appear in the lists. Define the following problem class mumdegree2,aswellasthoselistassignments(with for k, d ∈ IN : arbitrarily long lists) where each color occurs in at most two lists. (k, d)-LC : Sparse hard instances. The above examples show L that the following result is tight in several ways. Instance : Graph G =(V,E), list assignment = (L1,...,Ln), |Li| = k for all 1 ≤ i ≤ n, each color ap- Theorem 4.5. (Kratochv´ıl, Tuza [136]) The List pearing in at most d lists. Coloring NP problem is -complete when restricted to Question : Does L admit a list coloring on G ? the instances where each list contains at most 3 colors, each color occurs in at most 3 lists, and G is a planar Theorem 4.6. ([136]) Let k ≥ 3, d arbitrary. bipartite graph of maximum degree 3. (i) If d ≤ k, then every instance of (k, d)-LC admits a This result is proved by applying one of the several con- list coloring, and a feasible coloring can be found in Satisfiability nections between LC and the problem. O(n2.5)steps. Given a Boolean formula Φ in conjunctive normal form, with a set C of clauses over the set X of variables, one (ii) If d>k,then(k, d)-LC is NP-complete. can define a graph GΦ with vertex set V = X ∪ C and edge set The first part of this result means that the colorability does not depend on the actual structure of the graph in E := {xc x ∈ c ∈ C or ¬x ∈ c ∈ C} . question ; i.e., one may assume G = Kn.Inthiscase,a The symbols x and x (x ∈ X) will be taken for the colors, list coloring exists if and only if the lists admit distinct and the lists L(x)andL(c) for the variable vertices x and representatives, and therefore the problem is equally hard clause vertices c will be defined as as Bipartite Matching (or, more explicitly, as finding a matching that covers the smaller vertex class of a bipartite { }∀∈ L(x):= x, x x X graph). and Hall Condition. The following concept may be viewed L(c):={x x ∈ c}∪{x ¬x ∈ c}∀x ∈ X. as the LC-analogue of the Independence Condition given in Section 4.2. For graph G =(V,E), list assignment It can be seen that there is a one-to-one correspondence L, subset U ⊆ V , and color i ∈ IL , d e n o t e b y α(U, i) between the satisfying truth assignments of Φ and those the largest size of an independent set in the subgraph

20 induced by those vertices of U whose lists contain color i. and solvable in linear time for p = r ≥ q and for q ≤ p ≤ It is obvious that the condition q +1.  |U|≤ α(U, i) ∀ U ⊆ V Graphs of bounded treewidth. One of the equiv- i∈IL alent definitions of treewidth is introduced in terms of chordal graphs : is necessary for the existence of a list coloring. Hilton and Johnson [94] and Gr¨oflin [77] prove that this coindition is tw(G):=min{ω(H) − 1} , sufficient for all L on G if and only if each 2-connected component of G is a complete subgraph. In the particular where the minimum is taken over all chordal graphs H case of line graphs G = L(H), the necessary and suffi- containing G as a subgraph. Representing such an H as cient condition is that H should be a forest (de Werra the intersection graph of subtrees T1,...,Tn of a tree T , { ∈ ∈ } ∈ [48]). If multiple edges are also allowed, the Hall Condi- the sets Xz := vi V y V (Ti) (z V (T )) together tion becomes sufficient on multiforests if we require that with T form a so-called tree decomposition of G,avery any two parallel edges have the same list (Marcotte and convenient structure for algorithmic purposes. On this Seymour [145]). basis, for many NP-complete problems there exist poly- It may be noted that if all blocks are cliques, a polyno- nomial (and often even linear) algorithms when restricted mial list coloring algorithm can be designed even without to graphs of treewidth less than t (often called partial t- ∈ the above characterization at hand. For this, one can take trees), t IN fixed ; see e.g. [12, 44]. The methods of dynamical programming can be applied successfully for an endblock K sitting on a cut vertex vj ,andcheckfor list colorings as well : each color i ∈ Lj one by one whether ϕ(vj )=i can be ex- tended to a list coloring on the entire K. (This amounts Theorem 4.8. (Jansen, Scheffler [109]) Let t ∈ IN b e just to testing whether the L \{i} in K \{v } have j j fixed. Then, on the class G of graphs of treewidth distinct representatives.) Restricting L to those colors t j less than t,theList Coloring problem is solvable in which do, the problem gets reduced to the subgraph in- O(nt+2) time. Moreover, for every fixed k ∈ IN , i f a t r e e duced by V \ (V (K) \{v }) which is list colorable with j decomposition of width

21 solvable in polynomial time on Pn and also on the vertex- 4.5 Graph coloring games disjoint unions of paths. The case of |IL | =3wassolved previously by Xu [198]. Several games on graphs may be viewed as on-line versions of precoloring extension : At each step, the next player It remains open to investigate the complexity of the has to extend the partial coloring to a larger one. Here problem on trees, with a bounded total number of colors. we consider some two-person games of this flavor. In the variants below, it will be assumed throughout 4.4 Choosability that, already at the beginning of the game, both players While the hard instances of List Coloring turn out to know the entire graph G =(V,E) to be colored. More- be NP-complete, with respect to Choosability the class over, a color bound k is given. A legal move consists of Πp plays the role of NP. The 2-choosable graphs can be choosing a vertex v not colored so far, and assign to it an 2 ∈{ } recognized in linear time, by the structural characteri- arbitrary color i 1,...,k that has not been assigned zation (Theorem 3.3). Apart from this ‘ smallest ’ case, to any neighbor of v. We begin with a framework that essentially every other class of instances is provably hard. may be viewed as most general in some sense, and then The following result gives a complete answer to the prob- discuss some particular cases and variants. lem formulated at the beginning of this section. Achievement and Avoidance Games. In both types, the players move alternately, and the player to Theorem 4.9. (Gutner, Tarsi [83]) For every k ≥ 3, move next is obliged to color a vertex, whenever the k-Choosability is Πp-complete. 2 partial coloring admits an extension. The Achievement The first complexity result of this kind was due to by Game is won by the player who makes the last legal move ; Rubin [62], but not for lists of equal length. Recently, while in the Avoidance Game, the last-but-one move wins, Gutner proved several similar theorems. In order to state i.e., the winner is the player who can force the other one some of them, we need to introduce the following concept. to make the last move. Small values of k lead to some concepts interesting (2, 3)-Choosability ( (2, 3)-CH ) : on their own : For k =2bothgamesendupwithan inclusionwise maximal bipartite induced subgraph (with Instance : Graph G =(V,E), number  ∈{2, 3} for i unchangeable vertex 2-coloring in each of its components), each vertex v . i and for k = 1 they result in a nonextendable independent Question : Does G have a list coloring for every as- set. L | | signment =(L1,...,Ln) such that Li = i for all These games have been considered by Harary and ≤ ≤ 1 i n ? Tuza [92] for some rather restricted types of graphs G (paths, cycles, Petersen graph) with color bound k = Along these lines, a large class of problems χ(G). As may be expected, Avoidance turns out to be parametrized by sets S of natural numbers can also be more complicated than Achievement. Very little is known ∈ | | defined, assuming that i S and Li = i for each ver- so far in general, however, though it would be interesting L tex vi in the list assignment for which the colorability to see various winning strategies, as well as arguments has to be tested. With this formulation, Rubin’s theorem p showing that it is hard to determine the winner already states that (2, 3)-CH is Π2-complete on bipartite graphs. on some graph classes of a fairly transparent structure. Gutner proves the following stronger related results. For small k, the game is known to be PSPACE- Theorem 4.10. (Gutner [82]) Each of the following complete on unrestricted graphs, by the results of Schae- p fer (k = 1) and Bodlaender (k =2). problems is Π2-complete : (2, 3)-CH on planar bipartite graphs, Theorem 4.12. ([158], [20]) For color bounds k =1 3-CH on planar triangle-free graphs, and k =2,itisPSPACE-complete to decide who has a 4-CH on planar graphs, winning strategy in the Achievement Game. 3-CH on the union of two forests. Sofar,thecaseofk ≥ 3 colors seems to be open. On These results may raise the impression that choosability the other hand, more results are available under the con- is always at least as hard as list colorability. This is not at dition that the players have to color the vertices in a all the case, however, as shown by the comparison of the prescribed order. See Bodlaender [20] and Bodlaender next result with Theorem 4.7. We denote by (p, q, r)-CH and Kratsch [23] for details on those ‘ sequential color- thechoiceversionof(p, q, r)-LC. ing ’ games.

Theorem 4.11. (Kratochv´ıl, Tuza, Voigt [137]) If Symmetric strategies. The simplest example to illus- 2r ≥ p and 4q>3r+p, and also if 2r ≤ p and 4q>2p+r, trate the idea how the symmetry of a graph can be used then the (p, q, r)-CH problem is solvable in linear time. successfully, is the winning strategy of the first player in the Achievement Game on the path Pn, n odd. Denot- The complexity of (p, q, r)-CH, however, is not known in ing Pn = v0v1 ···v2t, Player 1 colors the middle vertex general. vt first (with any color), and then ‘ reflects ’ each move of

22 Player 2 to vt ; i.e., if Player 2 colors some vi with color j, Faigle, Kern, Kierstead and Trotter [64] proved then Player 1 assigns the same color j to the vertex v2t−i χg(T ) ≤ 4 for every tree T , and Bodlaender [20] showed in the next move. that this estimate is tight, by constructing a tree with In his recent work, Arroyo [13] applies this idea and χg =4.(LetT be the caterpillar with 4 internal nodes its modifications in designing winning strategies for the along a path, each of degree 4.) The upper bound has Achievement and/or Avoidance Games on various types been generalized by Kierstead and Tuza [127] who proved of graphs. Moreover, he considers several further vari- that ants of these games, e.g., where each player has to use χg(G) ≤ 6 tw(G) − 2 a prescribed set of colors (those sets may be disjoint for holds for every graph G,wheretw(G) denotes the the two players), or adjacent vertices must get the same treewidth of G (see the definition before Theorem 4.8). color, etc. It is not known, however, whether the coefficient 6 is re- Achievement for k =1(Node Kayles). The game ally necessary here, or it can be replaced by a smaller one with one color seems to be of major importance, because (with possibly a worse error term). the case of more colors can be reduced to it. Indeed, as It was conjectured by Bodlaender [20] and proved by Arroyo observes [13], the winner is the same on G with Kierstead and Trotter [126] that the game chromatic num- k colors and on the Cartesian product G2Kk with one ber of planar graphs is bounded above by a constant. The color. (The vertex set of G2Kk is V (G) ×{1,...,k},and largest possible value of χg, however, is known neither for two of its vertices (v, i)and(v,i) are adjacent if and only planar graphs (it is between 8 and 33), nor for outerplanar if i = i and vv ∈ E(G)ori = i and v = v.) graphs (between 6 and 8). For a general upper bound, we If just one color is available, the players sequentially recall the following result. construct larger and larger independent sets until a max- imal one is reached, and the first player wins if and only Theorem 4.13. (Kierstead, Trotter [126]) There exists if the set eventually obtained has odd cardinality. Be- a function g :IN→ IN such that, if a graph G does not side the complexity result mentioned above, Schaefer [158] contain any subgraph homeomorphic to Kt,thenχg(G) ≤ proves that the bipartite version of the game is PSPACE- g(t). complete, as well, i.e., where G is supposed to be bipar- tite, say with vertex partition V = V1 ∪ V2, and player i It follows, in particular, that the game chromatic number (i =1, 2) selects a vertex of Vi in each step. is bounded above by a function of the genus. Finbow and Hartnell [66] investigate, under which Though there is relatively little known about the be- conditions is the outcome of the game independent of the havior of the game chromatic number so far, it seems to actual strategies of the players, i.e., when are the maxi- offer a promising area for research, certainly with a lot mal independent sets of G all of the same parity. They more to discover. prove that for graphs of girth at least 8, the necessary and sufficient condition is that every vertex of degree greater than 1 is adjacent to an odd number of pendant vertices. Acknowledgements. I am indebted to N. Alon, (The girth condition cannot be weakened here, as shown J. A. Bondy, M. Hujter, T. R. Jensen, H. Kierstead, by the cycle C7.) J. Kratochv´ıl, P. Mih´ok, C. Thomassen, B. Toft, and The Achievement Game with k =1onpaths is dis- M. Voigt for fruitful and inspirative discussions on the cussed by Berlekamp, Conway and Guy [16, pp. 88– subject, to G. Bacs´o, K. M. Hangos, and T. R. Jensen for 90] in a different but equivalent form, under the name their many valuable comments on the first version of the ‘ Dawson’s Chess ’ (played on a 3 × n board with n white paper, and to N. Eaton, C. N. Jagger, and A.V. Kostochka pawns and n black pawns, initially placed in the first and for useful short remarks. I thank the authors of results third row, respectively ; capture is obligatory). Interest- cited here but not published so far, for kindly informing ingly enough, the score turns out to be ultimately periodic me about their most recent works. Also, support from modulo 34. The second player has a winning strategy on the Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin Pn if and only if n ≡ 4, 8, 20, 24, 28 (mod 34) or n =14 is thankfully acknowledged, where part of this research or n = 34. was carried out. Game chromatic number. This interesting concept was introduced by Bodlaender [20]. Depending on the References parity of n = |V (G)|, the game becomes some kind of Achievement (n odd) or avoidance (n even), but now the [1] M. Ajtai, J. Komlos,´ E. Szemeredi´ : Anoteon first player wins if and only if the entire graph gets col- Ramsey numbers. Journal of Combinatorial Theory, Ser. A 29 (1980), 354–360. ored. The game chromatic number of G, denoted χg(G), is the smallest integer k such that the first player wins [2] M. O. Albertson : You can’t paint yourself into a the game with color bound k.(Inordertoavoidsome corner . Manuscript, 1997. anomalies, Kierstead et al. propose a slight change in the [3] N. Alon : Choice numbers of graphs : A probabilistic rules, namely that Player 2 begins but he is allowed to approach. Combinatorics, Probability and Comput- pass.) ing 1 (1992), 107–114.

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