Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Xin Zhang, Guizhen Liu, and Jian-Liang Wu

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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

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

First available in Project Euclid: 6 December 2010

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Zentralblatt MATH identifier

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
Secondary: 05C15: Coloring of graphs and hypergraphs

Acyclic coloring frugal coloring $k$-forested coloring planar graphs girth


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.

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  • O. Amini, L. Esperet and J. van den Heuvel, Frugal Colouring of Graphs.
  • J. A. Bondy and U. S. R. Murty, Graph theory with applications, American Elsevier Publishing Co., Inc., New York, 1976.
  • O. V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979), no. 3, 211–236.
  • O. V. Borodin et al., Sufficient conditions for planar graphs to be 2-distance $(\Delta+1)$-colorable, Sib. Èlektron. Mat. Izv. 1 (2004), 129–141.
  • O. V. Borodin, A. O. Ivanova and T. K. Neustroeva, 2-distance coloring of sparse planar graphs, Sib. Èlektron. Mat. Izv. 1 (2004), 76–90.
  • O. V. Borodin, A. V. Kostochka and D. R. Woodall, Acyclic colourings of planar graphs with large girth, J. London Math. Soc. (2) 60 (1999), no. 2, 344–352.
  • T. F. Coleman and J.-Y. Cai, The cyclic coloring problem and estimation of sparse Hessian matrices, SIAM J. Algebraic Discrete Methods 7 (1986), no. 2, 221–235.
  • T. F. Coleman and J. J. Moré, Estimation of sparse Hessian matrices and graph coloring problems, Math. Programming 28 (1984), no. 3, 243–270.
  • L. Esperet, M. Montassier and A. Raspaud, Linear choosability of graphs, Discrete Math. 308 (2008), no. 17, 3938–3950.
  • B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (1973), 390–408.
  • H. Hind, M. Molloy and B. Reed, Colouring a graph frugally, Combinatorica 17 (1997), no. 4, 469–482.
  • H. Hind, M. Molloy and B. Reed, Total coloring with $\Delta+\text{poly}(\log\Delta)$ colors, SIAM J. Comput. 28 (1999), no. 3, 816–821.
  • A. Raspaud and W. Wang, Linear coloring of planar graphs with large girth, Discrete Math. 309 (2009), no. 18, 5678–5686.
  • R. Yuster, Linear coloring of graphs, Discrete Math. 185 (1998), no. 1–3, 293–297.