Equitable Vertex Arboricity Conjecture Holds for Graphs with Low Degeneracy

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 Δ is equitably tree- k -colorable for every integer k ≥ (Δ + 1)/2 provided that Δ ≥ 9.818 d , confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.