Abstract
The vertex arboricity $$\rho (G)$$?(G) of a graph $$G$$G is the minimum number of colors to color $$G$$G such that each color class induces a forest. The list vertex arboricity $$\rho _l(G)$$?l(G) is the list-coloring version of this concept. Zhen and Wu conjectured that $$\rho _l(G)=\rho (G)$$?l(G)=?(G) whenever $$|V(G)|\le 3\rho (G)$$|V(G)|≤3?(G). In this paper, we prove the weaker version of the conjecture obtained by replacing $$3\rho (G)$$3?(G) with $$\frac{5}{2}\rho (G)+\frac{1}{2}$$52?(G)+12.
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