Equitable vertex arboricity of $d$-degenerate graphs.
Xin Zhang, Bei Niu, Yan Li, Li Bi · arXiv (Cornell University) · 2019
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 $k\geq (\Delta+1)/2$.