Obstructions for homomorphisms to odd cycles in series-parallel graphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916662022766592 |
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| author | Cho, Eun-Kyung Choi, Ilkyoo Park, Boram Siggers, Mark |
| author_facet | Cho, Eun-Kyung Choi, Ilkyoo Park, Boram Siggers, Mark |
| contents | For a graph $H$, an $H$-colouring of a graph $G$ is a vertex map $ϕ:V(G) \to V(H)$ such that adjacent vertices are mapped to adjacent vertices. A graph $G$ is $C_{2k+1}$-critical if $G$ has no $C_{2k+1}$-colouring but every proper subgraph of $G$ has a $C_{2k+1}$-colouring. We prove a structural characterisation of $C_{2k+1}$-critical graphs when $k \geq 2$. In the case that $k = 2$, we use the aforementioned charazterisation to show a $C_3$-free series-parallel graph $G$ has a $C_5$-colouring if either $G$ has neither $C_8$ nor $C_{10}$, or $G$ has no two $5$-cycles sharing a vertex. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Obstructions for homomorphisms to odd cycles in series-parallel graphs Cho, Eun-Kyung Choi, Ilkyoo Park, Boram Siggers, Mark Combinatorics For a graph $H$, an $H$-colouring of a graph $G$ is a vertex map $ϕ:V(G) \to V(H)$ such that adjacent vertices are mapped to adjacent vertices. A graph $G$ is $C_{2k+1}$-critical if $G$ has no $C_{2k+1}$-colouring but every proper subgraph of $G$ has a $C_{2k+1}$-colouring. We prove a structural characterisation of $C_{2k+1}$-critical graphs when $k \geq 2$. In the case that $k = 2$, we use the aforementioned charazterisation to show a $C_3$-free series-parallel graph $G$ has a $C_5$-colouring if either $G$ has neither $C_8$ nor $C_{10}$, or $G$ has no two $5$-cycles sharing a vertex. |
| title | Obstructions for homomorphisms to odd cycles in series-parallel graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.19411 |