The list coloring and list total coloring of planar graphs with maximum degree at least 7

Abstract

A graph G is edge k-choosable (respectively, total k-choosable) if, whenever we are given a list L(x) of colors with |L(x)| = k for each x ∈ E(G) (x ∈ E(G) ∪ V (G)), we can choose a color from L(x) for each element x such that no two adjacent (or incident) elements receive the same color. The list edge chromatic index χ′l(G) (respectively, the list total chromatic number χ′′l(G)) of G is the smallest integer k such that G is edge (respectively, total) k-choosable. In this paper, we focus on a planar graph G, with maximum degree Δ (G) ≥ 7 and with some structural restrictions, satisfies χ′l(G) = Δ (G) and χ′′l(G) = Δ (G) + 1.