The List Point Arboricity of Some Complete Multi-partite Graphs

Abstract

Let $G$ be a graph. The point arboricity of $G$, denoted by $\rho (G)$, is the minimum number of colors that can be used to color the vertices of $G$ so that each color class induces an acyclic subgraph of $G$. The list point arboricity ${\rho }_l(G)$ is the minimum $k$ so that there is an acyclic $L$-coloring for any list assignment $L$ of $G$ which $|L(v)|\ge k$. So $\rho (G)\le {\rho }_l(G)$. Zhen and Wu conjectured that if $|V(G)|\le 3\rho (G)$, then ${\rho }_l(G)=p(G)$. Motivated by this, we investigate the list point arboricity of some complete multi-partite graphs of order slightly larger than $3p(G)$, and obtain $\rho (K_{m,(1),2(n-1)})={\rho }_l(K_{m(1),2(n-1)})$ $(m=2,3,4)$.