The square of a subcubic planar graph without a 5-cycle is 7-choosable

Abstract: The square of a graph $G$, denoted $G<sup>2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen [12] showed that $χ(G²⁾ \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ<em>{\ell}(G²⁾ \leq 7$ or not if $G$ is a subcubic planar graph. Recently Kim and Lian [11] showed that $χ</em>{\ell}(G²⁾ \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that $χ_{\ell}(G²⁾ \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles. In this paper, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the results of [10] and [11].