  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.
