Choice number of 3-colorable elementary graphs

We show that the choice number of a graph G is equal to its chromatic number when G belongs to a restricted class of claw-free graphs, in view of the conjecture that this is true for every claw-free graph. We consider only finite, undirected graphs, without loops. Given a graph G = ( V, E), a k-coloring of the vertices of G is a mapping c : V + { 1,2,. . , k} such that for every edge my of G we have c(x) # c(y). The graph G is called k-colorable if it admits a k-coloring, and x(G) denotes the smallest of all integers k such that G is k-colorable. ErdGs et al. [2] introduced a variant of the coloring problem as follows. Suppose that each vertex u is assigned a list L(V) of possible colors; we then want to find a vertex-coloring c such that c(v) t L(u) for all 2: E V. When such a c exists we will say that the graph G is L-colorable; we may also say that c is an L-coloring of G. Given an integer k, the graph G is called k-choosahle if it is L-colorable for every assignment L that satisfies IL(v)l = k for all v E V. Finally, the choice number or listchromatic number Ch(G) of G is the smallest k such that G is k-choosable. Clearly every k-choosable graph is k-colorable (consider the assignment L(V) = { 1,2.. , k} for all ZJ E V), and so holds for every graph. We would like to call this the choice chromatic inequality. The converse inequality however is not true; in fact there exist bipartite graphs with arbitrarily high choice number. When coloring the edges of a graph rather than the vertices, the above notions are extended in the obvious way. The least number of colors necessary to color all edges * Corresponding author. E-mail: frederic.maffray@imag.fr 0012-365X/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PII SOOl2-365X(96)00182-3 354 S. Gravier, F. Maffiayl Discrete Mathematics 1651166 (1997) 353-358 of a graph with no two incident edges receiving the same color is its chromatic index x’(G). The least k such that G is L’-edge-colorable for any assignment L’ of colors to the edges with IL’(e)/ = k for all eEE is called the choice index or list-chromatic index of G. A famous conjecture, attributed in particular to Vizing (see [5]), is: Conjecture 1 (List chromatic conjecture). Every graph G satisfies Ch’(G) = x’(G). The specialized case of this conjecture dealing with complete bipartite graphs was known as the Dinitz conjecture, as it was equivalent to a problem on Latin squares posed by Jeffrey Dinitz. This is now a consequence of the following more general result due to Galvin [3]. Theorem 1 (Galvin [3]). Every bipartite graph G satisfies Ch’(G) = x’(G) The problem of edge-coloring can be reduced to a special instance of the problem of vertex-coloring via the line-graph. Recall that given a graph H, its line-graph 9(H) is the graph whose vertices are the edges of H and whose edges are the pairs of incident edges of H. It is clear that x(Z(H))=f(H) and Ch(Z(H))=Ch’(H). Now a reformulation of the above theorem is that the choice number of the line-graph of any bipartite graph is equal to its chromatic number, and a reformulation of Conjecture 1 is that every line-graph should satisfy the choice chromatic equality. The bipartite graph K~,J is usually called the claw, and any graph that does not contain an induced claw is called claw-free. It is easy to see that every line-graph is claw-free. Previously the choice chromatic equality was established for a couple of restricted classes of claw-free graphs, in particular for complements of trianglefree graphs [4]. Here we are concerned with the class of claw-free graphs known as elementary graphs, defined by Chvkal and Sbihi [l]. A graph is called elementary if its edges can be colored with two colors such that every P3 (chordless path on three vertices) has its two edges colored differently. Clearly elementary graphs are claw-free. The structure of elementary graphs was determined in [6] as follows. An edge is called Jat if it does not lie in a triangle. Let my be a flat edge of a graph G and (X, Y;F) a cobipartite graph disjoint from G, where X, Y are disjoint non-empty cliques and there exists at least one edge from X to Y. We can obtain a new graph from G U (X, Y;F) by removing x and y and adding all the edges between X and N(x) y and all edges between Y and N(y)-x. This is called augmenting the flat edge xy with the cobipartite graph (X, Y; F) (see Fig. 1). More generally, consider h pairwise non-incident flat edges ei,. . . ,eh of G and h co-bipartite graphs (xi, Yl; El), . . . ,(&, Yh;&) that are mutually disjoint and disjoint from G, such that X;:, K are two disjoint non-empty cliques, and there exists at least one edge from Xi to K. We can augment simultaneously the edges ei, . . . . eh with (Xl,Yl;El),...,(Xh,Yh;Eh), respectively. Since these h edges are nonincident, the graph that results from these augmentations is the same regardless of the order in which they are performed, and is called an augmentation of G. The cobipartite graphs (Xi, K; Ei) are called the augments. S. Grauier, F. Maffray I Discrete Mathematics 1651166 (1997) 353-358 355