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December 2010 $k$-forested coloring of planar graphs with large girth
Xin Zhang, Guizhen Liu, Jian-Liang Wu
Proc. Japan Acad. Ser. A Math. Sci. 86(10): 169-173 (December 2010). DOI: 10.3792/pjaa.86.169


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 forest with maximum degree at most $k$. The $k$-forested chromatic number of a graph $G$, denoted by $\chi^{a}_{k}(G)$, is the smallest number of colors in a $k$-forested coloring of $G$. In this paper, it is shown that planar graphs with large enough girth do satisfy $\chi^{a}_{k}(G)=\lceil\frac{\Delta(G)}{k}\rceil+1$ for all $\Delta(G)> k\geq 2$, and $\chi^{a}_{k}(G)\leq 3$ for all $\Delta(G)\leq k$ with the bound 3 being sharp. Furthermore, a conjecture on $k$-frugal chromatic number raised in [1] has been partially confirmed.


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Xin Zhang. Guizhen Liu. Jian-Liang Wu. "$k$-forested coloring of planar graphs with large girth." Proc. Japan Acad. Ser. A Math. Sci. 86 (10) 169 - 173, December 2010.


Published: December 2010
First available in Project Euclid: 6 December 2010

zbMATH: 1208.05022
MathSciNet: MR2779830
Digital Object Identifier: 10.3792/pjaa.86.169

Primary: 05C10
Secondary: 05C15

Keywords: $k$-forested coloring , Acyclic coloring , frugal coloring , Girth , Planar graphs

Rights: Copyright © 2010 The Japan Academy

Vol.86 • No. 10 • December 2010
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