## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### $k$-forested coloring of planar graphs with large girth

#### 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 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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 10 (2010), 169-173.

Dates
First available in Project Euclid: 6 December 2010

https://projecteuclid.org/euclid.pja/1291644507

Digital Object Identifier
doi:10.3792/pjaa.86.169

Mathematical Reviews number (MathSciNet)
MR2779830

Zentralblatt MATH identifier
1208.05022

#### Citation

Zhang, Xin; Liu, Guizhen; Wu, Jian-Liang. $k$-forested coloring of planar graphs with large girth. Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 10, 169--173. doi:10.3792/pjaa.86.169. https://projecteuclid.org/euclid.pja/1291644507

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