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 tree-k-coloring of graph is vertex coloring using k distinct colors such that every color class induces forest and sizes any two classes differ by at most one. In this paper, we show theoretical results graphs proving d-degenerate maximum degree ? equitably tree-k-colorable for integer ? (? + 1)/2 provided 9.818d, confirming arboricity conjecture low degeneracy.