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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.20087 |
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| _version_ | 1866916810687774720 |
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| author | Lo, On-Hei Solomon |
| author_facet | Lo, On-Hei Solomon |
| contents | A theorem of Tutte states that every 4-connected non-hamiltonian graph contains $K_{3,3}$ as a minor. We strengthen this result by proving that such a graph must contain $K_{3,4}$ as a minor, thereby confirming a special case of a conjecture posed by Chen, Yu, and Zang in a strong form. This result may be viewed as a step toward characterizing the minor-minimal 4-connected non-hamiltonian graphs. As a 3-connected analog, Ding and Marshall conjectured that every 3-connected non-hamiltonian graph has a minor of $K_{3,4}$, $\mathfrak{Q}^+$, or the Herschel graph, where $\mathfrak{Q}^+$ is obtained from the cube by adding a new vertex adjacent to three independent vertices. We confirm this conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20087 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On minors of non-hamiltonian graphs Lo, On-Hei Solomon Combinatorics A theorem of Tutte states that every 4-connected non-hamiltonian graph contains $K_{3,3}$ as a minor. We strengthen this result by proving that such a graph must contain $K_{3,4}$ as a minor, thereby confirming a special case of a conjecture posed by Chen, Yu, and Zang in a strong form. This result may be viewed as a step toward characterizing the minor-minimal 4-connected non-hamiltonian graphs. As a 3-connected analog, Ding and Marshall conjectured that every 3-connected non-hamiltonian graph has a minor of $K_{3,4}$, $\mathfrak{Q}^+$, or the Herschel graph, where $\mathfrak{Q}^+$ is obtained from the cube by adding a new vertex adjacent to three independent vertices. We confirm this conjecture. |
| title | On minors of non-hamiltonian graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.20087 |