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arXiv · 2512.24536

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

Seog-Jin Kim·Xiaopan Lian·Atsuhiro Nakamoto·Kenta Ozeki
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Abstract

The square of a graph $G$, denoted $G^2$, 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 $\chi(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently Kim and Lian [11] showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that $\chi_{\ell}(G^2) \leq 7$ 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].

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Seog-Jin Kim, Xiaopan Lian, Atsuhiro Nakamoto, Kenta Ozeki. 2025-12-31. The square of a subcubic planar graph without a 5-cycle is 7-choosable. https://arxiv.org/abs/2512.24536

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