Sufficient conditions for a planar graph to be list edge Δ-colorable and list totally (Δ + 1) -colorable

摘要

Let χl′(G), χl″(G) and Δ(G) denote the list chromatic index, the list total chromatic number and the maximum degree of a graph G, respectively. In this paper, we show that χl′(G)=Δ(G) and χl″(G)=Δ(G)+1 if G is a planar graph with Δ(G)≥7 and without adjacent cycles of length at most 4; ora planar graph with Δ(G)≥8 and without cycles of length 3 adjacent to cycles of length 5. Moreover, χl′(G)≤Δ(G)+1 and χl″(G)≤Δ(G)+2 if G is a planar graph with Δ(G)≥6 and without cycles of length 3 adjacent to cycles of length 5.