The List Point Arboricity of Some Complete Multi-partite Graphs

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 vertical bar L(v)vertical bar >= k. So rho(G) <= rho(l)(G). Zhen and Wu conjectured that if vertical bar V(G)vertical bar <= 3 rho(G), then rho(l)(G) = rho(G). Motivated by this, we investigate the list point arboricity of some complete multi-partite graphs of order slightly larger than 3 rho(G), and obtain p(K-m(1),K- 2(n - 1)) = rho(l)(K-m(1),K- 2(n - 1)) (m = 2, 3, 4).