Elsevier

Discrete Mathematics

Volume 312, Issue 5, 6 March 2012, Pages 999-1005
Discrete Mathematics

k-forested choosability of planar graphs and sparse graphs

https://doi.org/10.1016/j.disc.2011.10.031Get rights and content
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Abstract

A proper vertex coloring of a simple graph G is k-forested if the subgraph induced by the vertices of any two color classes is a k-forest, i.e. a forest with maximum degree at most k. The 2-forested coloring is also known as linear coloring, which has been extensively studied in the literature. In this paper, we aim to extend the study of 2-forested coloring to general k-forested colorings for every k3 by combinatorial means. Precisely, we prove that for a fixed integer k3 and a planar graph G with maximum degree Δ(G)Δ and girth g(G)g, if (Δ,g){(k+1,10),(2k+1,8),(4k+1,7)}, then the k-forested chromatic number of G is actually Δ(G)k+1. Moreover, we also prove that the k-forested chromatic number of a planar graph G with maximum degree Δ(G)k+1 and girth g(G)8 is Δ(G)k+1 provided k7. In addition, we show that the k-forested chromatic number of an outerplanar graph G is at most Δ(G)k+2 for every k2. In fact, all these results are proved for not only planar graphs but also for sparse graphs, i.e. graphs having a low maximum average degree mad(G), and we actually prove a choosability version of these results.

Keywords

k-forested coloring
Linear coloring
Frugal coloring
Maximum average degree
Planar graph

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This research is supported in part by the GIIFSDU grant yzc-10040, the RFDP grant 20100131120017 and the NSFC grants 10971121, 11101243, 61070230.