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arXiv · 1403.5013

Tree-colorable maximal planar graphs

Enqiang Zhu·Zepeng Li·Zehui Shao·Jin Xu
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Abstract

A tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graph $G$ with $δ(G)\geq 4$ contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles.

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Enqiang Zhu, Zepeng Li, Zehui Shao, Jin Xu. 2014-03-20. Tree-colorable maximal planar graphs. https://arxiv.org/abs/1403.5013

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