Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that χl″(G)≤Δ(G)+2 if Δ(G)≥6, and χl′(G)=Δ(G) and χl″(G)=Δ(G)+1 if Δ(G)≥8, where χl′(G) and χl″(G) denote the list edge chromatic number and list total chromatic number of G, respectively.

► The list edge colorings and list total colorings of planar graphs without triangles adjacent 4-cycles are investigated. ► We proved that, if a planar graph G without triangles adjacent 4-cycles and Δ(G)≥8, then χl′(G)=Δ(G). ► It is proved that χl″(G)≤Δ(G)+2, where G is a planar graph without triangles adjacent 4-cycles and Δ(G)≥6. ► It is proved that χl″(G)=Δ(G)+1, where G is a planar graph without triangles adjacent 4-cycles and Δ(G)≥8.