The square of every subcubic planar graph without 4-cycles and 5-cycles is 7-choosable
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| Format: | Preprint |
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2025
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| _version_ | 1866911030433546240 |
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| author | Jin, Ligang Kang, Yingli Kim, Seog-Jin |
| author_facet | Jin, Ligang Kang, Yingli Kim, Seog-Jin |
| contents | The square of a graph $G$, denoted by $G^2$, 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 (2018) and independently, Hartke, Jahanbekam and Thomas (2016) proved that $χ(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently, Kim and Lian (2024) proved that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. In this paper, we prove that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles, which improves the result of Kim and Lian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00426 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The square of every subcubic planar graph without 4-cycles and 5-cycles is 7-choosable Jin, Ligang Kang, Yingli Kim, Seog-Jin Combinatorics The square of a graph $G$, denoted by $G^2$, 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 (2018) and independently, Hartke, Jahanbekam and Thomas (2016) proved that $χ(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently, Kim and Lian (2024) proved that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. In this paper, we prove that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles, which improves the result of Kim and Lian. |
| title | The square of every subcubic planar graph without 4-cycles and 5-cycles is 7-choosable |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.00426 |