Author/Authors :
Zhang H. نويسنده School of Mathematical Science, Huaiyin Normal University, 111 Changjiang West Road, Huaian, Jiangsu, 223300, P. R. China.
Abstract :
The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph. A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$, one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by every color class is a forest. The smallest $k$ for a graph to be list vertex $k$-arborable is denoted by $rho_l(G)$. Borodin, Kostochka and Toft (Discrete Math. 214 (2000) 101-112) first introduced the list vertex arboricity of $G$. In this paper, we prove that $rho_l(G)leq 2$ for any toroidal graph without 5-cycles. We also show that $rho_l(G)leq 2$ if $G$ contains neither adjacent 3-cycles nor cycles of lengths 6 and 7.
