Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916053505802240 |
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| author | Carter, Daniel Yip, Jung Hon |
| author_facet | Carter, Daniel Yip, Jung Hon |
| contents | We prove Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs, where the co-claw is the disjoint union of a triangle and a vertex, the co-gem is the disjoint union of a 4-vertex path and a vertex, the fork is obtained from $K_{1,3}$ by subdividing one of the edges, and the antifork is the complement of the fork. The $\{\text{co-claw}, \text{co-gem}\}$-free graphs include the complements of line graphs of triangle-free multigraphs, and thus our results imply Hadwiger's Conjecture for these graphs. In fact, we prove a stronger result: every $\{\text{co-claw}, \text{co-gem}\}$-free graph $G$ has a $K_{χ(G)}$-model where each branch set has size at most 2, and every $\{\text{fork}, \text{antifork}\}$-free graph $G$ has a $K_{χ(G)}$-model where at most one branch set has size greater than 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28050 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs Carter, Daniel Yip, Jung Hon Combinatorics 05C83 (Primary), 05C15, 05C35 We prove Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs, where the co-claw is the disjoint union of a triangle and a vertex, the co-gem is the disjoint union of a 4-vertex path and a vertex, the fork is obtained from $K_{1,3}$ by subdividing one of the edges, and the antifork is the complement of the fork. The $\{\text{co-claw}, \text{co-gem}\}$-free graphs include the complements of line graphs of triangle-free multigraphs, and thus our results imply Hadwiger's Conjecture for these graphs. In fact, we prove a stronger result: every $\{\text{co-claw}, \text{co-gem}\}$-free graph $G$ has a $K_{χ(G)}$-model where each branch set has size at most 2, and every $\{\text{fork}, \text{antifork}\}$-free graph $G$ has a $K_{χ(G)}$-model where at most one branch set has size greater than 2. |
| title | Hadwiger's Conjecture for $\{\text{co-claw}, \text{co-gem}\}$-free graphs and $\{\text{fork}, \text{antifork}\}$-free graphs |
| topic | Combinatorics 05C83 (Primary), 05C15, 05C35 |
| url | https://arxiv.org/abs/2605.28050 |