A minimization problem in graph theory so-called the equitable tree-coloring problem can be used to formulate a structure decomposition problem on the communication network with some security considerations. Precisely, 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 establish some theoretical results on the equitable tree-colorings of graphs by showing 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 $\Delta\geq 10d$. This generalises the result of Chen et al.[J. Comb. Optim. 34(2) (2017) 426--432] which states that every $5$-degenerate graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$.

中文翻译：

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$d$-退化图的公平顶点树状

图论中的最小化问题，即所谓的公平树着色问题，可用于在具有某些安全考虑的通信网络上制定结构分解问题。准确地说，图的一个 equitable tree-$k$-coloring 是使用 $k$ 不同颜色的顶点着色，这样每个颜色类别都会导致森林，并且任何两个颜色类别的大小最多相差一个。在本文中，我们通过证明具有最大度数至多 $\Delta$ 的每个 $d$-退化图对于每个整数 $k 都是公平树 $k$-可着色的，从而建立了一些关于图的公平树着色的理论结果\geq (\Delta+1)/2$ 假设 $\Delta\geq 10d$。这概括了 Chen 等人的结果。 [J. 梳子。优化。