List Point Arboricity of Dense Graphs

Abstract

Let G be a simple graph. The point arboricity ρ(G) of G is defined as the minimum number of subsets in a partition of the point set of G so that each subset induces an acyclic subgraph. The list point arboricity ρ _l (G) is the minimum k so that there is an acyclic L-coloring for any list assignment L of G which |L(v)| ≥ k. So ρ(G) ≤ ρ _l (G) for any graph G. Xue and Wu proved that the list point arboricity of bipartite graphs can be arbitrarily large. As an analogue to the well-known theorem of Ohba for list chromatic number, we obtain ρ _l (G + K _n ) = ρ(G + K _n ) for any fixed graph G when n is sufficiently large. As a consequence, if ρ(G) is close enough to half of the number of vertices in G, then ρ _l (G) = ρ(G). Particularly, we determine that $\rho_l(K_{2(n)})=\lceil \frac {2n}{3}\rceil$, where K _2(n) is the complete n-partite graph with each partite set containing exactly two vertices. We also conjecture that for a graph G with n vertices, if $\rho(G)\geq \frac {n} {3}$ then ρ _l (G) = ρ(G).