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

Abstract

The First-Fit (or Grundy) chromatic number of a graph G denoted by $$\chi _{{_\mathsf{FF}}}(G)$$źFF(G), is the maximum number of colors used by the First-Fit (greedy) coloring algorithm when applied to G. In this paper we first show that any graph G contains a bipartite subgraph of Grundy number $$\lfloor \chi _{{_\mathsf{FF}}}(G) /2 \rfloor +1$$źźFF(G)/2ź+1. Using this result we prove that for every $$t\ge 2$$tź2 there exists a real number $$c>0$$c>0 such that in every graph G on n vertices and without cycles of length 2t, any First-Fit coloring of G uses at most $$cn^{1/t}$$cn1/t colors. It is noted that for $$t=2$$t=2 this bound is the best possible. A compactness conjecture is also proposed concerning the First-Fit chromatic number involving the even girth of graphs.