List Point Arboricity Of Total Graphs And List Coloring Of Planar Graphs

Firstly, we investigate the list point arboricity of total graphs, and conjecturethat [(Δ(G)+1)/2]≤ρ(T(G)) =ρ_l(T(G))≤[(Δ(G)+2/2] for any graph G. We show that[(Δ(G)+1)/2]≤ρ(T (G))≤ρ_l(T (G))≤[(Δ(G)+2)/2]for a 2-degenerate graph G. In particular,if G is an outerplanar graph with ?(G) = 3 , thenρ(T(G)) =ρ_l(T(G)) = [(Δ(G)+1)/2]unless G =|～P₂.Secondly, we investigate the list coloring of planar graphs, and give a su?cientcondition for a planar graph being 3-choosable. A graph G = (V,E) is L-colorable iffor a given list assignment L = {L(v) : v∈V (G)}, there exists a proper coloring c ofG such that c(v)∈L(v) for all v∈V . If G is L-colorable for every list assignment Lwith |L(v)|≥k for all v∈V , then G is said to be k-choosable. We prove that everyplanar graph without 4-,5-,7-cycles, and without triangles at distance less than 2is 3-choosable.