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arXiv · 1908.05066

Equitable vertex arboricity conjecture holds for graphs with low degeneracy

Xin Zhang·Bei Niu·Yan Li·Bi Li
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Abstract

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every $d$-degenerate graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$ provided that $\Delta\geq 9.818d$, confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.

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Xin Zhang, Bei Niu, Yan Li, Bi Li. 2019-08-14. Equitable vertex arboricity conjecture holds for graphs with low degeneracy. https://arxiv.org/abs/1908.05066

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