On the structure of (dart, odd hole)-free graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915265675001856 |
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| author | Hoàng, Chính T. |
| author_facet | Hoàng, Chính T. |
| contents | A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in the literature. We prove that a (dart, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or is the join or co-join of two smaller graphs. Using this structure result, we design a polynomial-time algorithm for finding an optimal colouring of (dart, odd hole)-free graphs. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $ω^2$ where $ω$ denotes the number of vertices in a largest clique of $G$. We prove that (dart, odd hole)-free graphs are perfectly divisible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_20422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the structure of (dart, odd hole)-free graphs Hoàng, Chính T. Combinatorics Discrete Mathematics 05C15, 05C85 A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in the literature. We prove that a (dart, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or is the join or co-join of two smaller graphs. Using this structure result, we design a polynomial-time algorithm for finding an optimal colouring of (dart, odd hole)-free graphs. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $ω^2$ where $ω$ denotes the number of vertices in a largest clique of $G$. We prove that (dart, odd hole)-free graphs are perfectly divisible. |
| title | On the structure of (dart, odd hole)-free graphs |
| topic | Combinatorics Discrete Mathematics 05C15, 05C85 |
| url | https://arxiv.org/abs/2504.20422 |