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 \(|L(v)| \geq k\). So \(\rho(G) \leq \rho_l(G)\). Zhen and Wu conjectured that if \(|V(G)| \leq 3\rho (G)\), 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 \(\rho(K_{m,(1),2(n-1)}) = \rho_l(K_{m(1),2(n-1)})\) \((m = 2,3,4)\).