Results on the Grundy chromatic number of graphs Original Research Article

Given a graph G, by a Grundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, image, every vertex of G colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by image and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining image, for any fixed integer k and show that it is a polynomial time problem. But in general, Grundy number is an NP-complete problem. We show that it is NP-complete even for the complement of bipartite graphs and describe the Grundy number of these graphs in terms of the minimum edge dominating number of their complements. Next we obtain some additive Nordhaus–Gaddum-type inequalities concerning image and image, for a few family of graphs. We introduce well-colored graphs, which are graphs G for which applying every greedy coloring results in a coloring of G with image colors. Equivalently G is well colored if image. We prove that the recognition problem of well-colored graphs is a coNP-complete problem.