κ-forested coloring of planar graphs with large girth

Abstract

A proper vertex coloring of a simple graph G is κ-forested if the subgraph induced by the vertices of any two color classesisa forest with maximum degree at mostk. Thek-forested chromatic number of a graph G, denoted by Xak(G), is the smallest number of colors in a κ-forested coloring of G. In this paper, it is shown that planar graphs with large enough girth do satisfy Xak(G) = ⌈Δ(G)/κ + 1 for all Δ(G) ≥ 2, and Xak(G) ≤ 3 for all Δ(G) ≤ κ with the bound 3 being sharp. Furthermore, a conjecture on k-frugal chromatic number raised in [1] has been partially confirmed. © 2010 The Japan Academy.