On Structure of Some Plane Graphs with Application to Choosability

A graph G=(V, E) is (x, y)-choosable for integers x>y⩾1 if for any given family {A(v) ∣ v∈V} of sets A(v) of cardinality x, there exists a collection {B(v) ∣ v∈V} of subsets B(v)⊂A(v) of cardinality y such that B(u)∩B(v)=∅ whenever uv∈E(G). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if G is free of k-cycles for some k∈{3, 4, 5, 6}, or if any two triangles in G have distance at least 2, then G is (4m, m)-choosable for all nonnegative integers m. When m=1, (4m, m)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable.