Results on the Grundy chromatic number of graphs

Abstract

The Grundy number of a graph G is the maximum number of colors used by the First-Fit coloring of G and is denoted by Γ(G). Similarly, the bchromatic number b(G) of G expresses the worst case behavior of another well-known coloring procedure i.e. color-dominating coloring of G. We obtain some families of graphs F for which there exists a function f (x) such that Γ(G) ≤ f (b(G)), for each graph G from the family. Call any such family (Γ, b)-bounded family. We conjecture that the family of b-monotone graphs is (Γ, b)-bounded and validate the conjecture for some families of graphs.

Dynamic F-free Coloring of Graphs

“…For proper coloring the largest number of colors used by the greedy algorithm is a well known graph invariant, denoted by Γ(G) and called the Grundy number of a graph. In this sense, Goyal and Vishwanathan [20], and Zaker [28] determined the computational complexity of a long-standing open problem posed by Hedetniemi et al [22] (cf. Jensen and Toft [25]).…”

First-Fit colorings of graphs with no cycles of a prescribed even length

“…(Gyárfás and Lehel 1988;Kierstead et al 1995;Kierstead and Trotter 1991). The First-Fit coloring in the form of an off-line coloring is called Grundy coloring (Zaker 2006). We begin with some necessary definitions.…”

“…, k} such that for any two colors i and j with i < j, any vertex colored j is adjacent to some vertex colored i. The Grundy or First-Fit chromatic number of a graph G, denoted by χ FF (G) (also denoted by (G) in some articles), is the largest integer k, such that there exists a Grundy k-coloring for G. The Grundy number and First-Fit coloring of graphs were studied widely in the literature, see Zaker (2006) and its references. We note that χ FF (G) equals to the maximum number of colors used by the on-line First-Fit coloring of G (Zaker 2006).…”

New bounds for the chromatic number of graphs

“…and T l,t , where 1 ≤ l ≤ t + 1. As pointed out in [13], any tree T with (T ) = t has at least 2 t−1 vertices, and up to isomorphism there is only one tree with this minimum order. Let's call it T t .…”

“…In Section 3 we first report a result to show the behavior of the greedy coloring algorithm on graphs with high girth. This is done in terms of Grundy coloring of graphs which is in fact the off-line version of greedy colorings (or First-Fit colorings) [13]. Next, in Section 3 we give a bound of the order of magnitude O(n 1− ) for the coloring number of graphs where n is the order, and some bounds for the chromatic number in terms of order and book size.…”