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

Abstract

Let $G$ be a planar graph with no two 3-cycles sharing an edge. We show that if $Δ(G)\geq 9$, then $χ'_l(G) = Δ(G)$ and $χ''_l(G)=Δ(G)+1.$ We also show that if $Δ(G)\geq 6$, then $χ'_l(G)\leqΔ(G)+1$ and if $Δ(G)\geq 7$, then $χ''_l(G)\leqΔ(G)+2$. All of these results extend to graphs in the projective plane and when $Δ(G)\geq 7$ the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if $G$ is a planar graph with no two 3-cycles sharing an edge and with $Δ(G)\geq 7$, then $G$ has an edge $uv$ with $d(u)\leq 4$ and $d(u)+d(v)\leq Δ(G)+2$. All of our proofs yield linear-time algorithms that produce the desired colorings.