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.

翻译：公平的树彩色可以在通信网络上提出结构分解问题, 并带有一些安全考虑。 也就是说, 图表的公平树- k$- 彩色是使用美元不同的颜色的顶点颜色, 使每个颜色类别诱发森林, 任何两个颜色类别的规模在最大程度上各有不同。 在本文中, 我们通过证明每张最大程度为$\ Delta$的美元- 下色图在每张整数为$k/ geq (\ Delta+1) 时, 都具有公平的树- k$- 彩色(\ Delta+1) /2$, 条件是每张整数为$\ Delta\ geq 9. 818d 美元, 以确认低降色度的图形的公平顶点偏角偏角的理论结果 。