Planar graphs without 5-cycles or without 6-cycles

Abstract

Let $G$ be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if $G$ is connected and $\delta \left(G\right)\ge 2$, then there exists an edge $xy\in E\left(G\right)$ such that $d\left(x\right)+d\left(y\right)\le 9$, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity $l{a}_2\left(G\right)\le \lceil \frac{\Delta \left(G\right)+1}{2}\rceil +6$, (2) its list total chromatic number is $\Delta \left(G\right)+1$ if $\Delta \left(G\right)\ge 8$, and (3) its list edge chromatic number is $\Delta \left(G\right)$ if $\Delta \left(G\right)\ge 8$.