Minimal obstructions to $C_5$-coloring in hereditary graph classes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910060874039296 |
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| author | Goedgebeur, Jan Jooken, Jorik Okrasa, Karolina Rzążewski, Paweł Schaudt, Oliver |
| author_facet | Goedgebeur, Jan Jooken, Jorik Okrasa, Karolina Rzążewski, Paweł Schaudt, Oliver |
| contents | For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Note that if $H$ is the triangle, then $H$-colorings are equivalent to $3$-colorings. In this paper we are interested in the case that $H$ is the five-vertex cycle $C_5$.
A minimal obstruction to $C_5$-coloring is a graph that does not have a $C_5$-coloring, but every proper induced subgraph thereof has a $C_5$-coloring. In this paper we are interested in minimal obstructions to $C_5$-coloring in $F$-free graphs, i.e., graphs that exclude some fixed graph $F$ as an induced subgraph. Let $P_t$ denote the path on $t$ vertices, and let $S_{a,b,c}$ denote the graph obtained from paths $P_{a+1},P_{b+1},P_{c+1}$ by identifying one of their endvertices.
We show that there is only a finite number of minimal obstructions to $C_5$-coloring among $F$-free graphs, where $F \in \{ P_8, S_{2,2,1}, S_{3,1,1}\}$ and explicitly determine all such obstructions. This extends the results of Kamiński and Pstrucha [Discr. Appl. Math. 261, 2019] who proved that there is only a finite number of $P_7$-free minimal obstructions to $C_5$-coloring, and of Dębski et al. [ISAAC 2022 Proc.] who showed that the triangle is the unique $S_{2,1,1}$-free minimal obstruction to $C_5$-coloring.
We complement our results with a construction of an infinite family of minimal obstructions to $C_5$-coloring, which are simultaneously $P_{13}$-free and $S_{2,2,2}$-free. We also discuss infinite families of $F$-free minimal obstructions to $H$-coloring for other graphs $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal obstructions to $C_5$-coloring in hereditary graph classes Goedgebeur, Jan Jooken, Jorik Okrasa, Karolina Rzążewski, Paweł Schaudt, Oliver Combinatorics 05C15, 05C85, 68W05, 68R10 For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Note that if $H$ is the triangle, then $H$-colorings are equivalent to $3$-colorings. In this paper we are interested in the case that $H$ is the five-vertex cycle $C_5$. A minimal obstruction to $C_5$-coloring is a graph that does not have a $C_5$-coloring, but every proper induced subgraph thereof has a $C_5$-coloring. In this paper we are interested in minimal obstructions to $C_5$-coloring in $F$-free graphs, i.e., graphs that exclude some fixed graph $F$ as an induced subgraph. Let $P_t$ denote the path on $t$ vertices, and let $S_{a,b,c}$ denote the graph obtained from paths $P_{a+1},P_{b+1},P_{c+1}$ by identifying one of their endvertices. We show that there is only a finite number of minimal obstructions to $C_5$-coloring among $F$-free graphs, where $F \in \{ P_8, S_{2,2,1}, S_{3,1,1}\}$ and explicitly determine all such obstructions. This extends the results of Kamiński and Pstrucha [Discr. Appl. Math. 261, 2019] who proved that there is only a finite number of $P_7$-free minimal obstructions to $C_5$-coloring, and of Dębski et al. [ISAAC 2022 Proc.] who showed that the triangle is the unique $S_{2,1,1}$-free minimal obstruction to $C_5$-coloring. We complement our results with a construction of an infinite family of minimal obstructions to $C_5$-coloring, which are simultaneously $P_{13}$-free and $S_{2,2,2}$-free. We also discuss infinite families of $F$-free minimal obstructions to $H$-coloring for other graphs $H$. |
| title | Minimal obstructions to $C_5$-coloring in hereditary graph classes |
| topic | Combinatorics 05C15, 05C85, 68W05, 68R10 |
| url | https://arxiv.org/abs/2404.11704 |