Totally odd immersions of complete graphs in graph products
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918212832067584 |
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| author | Echeverría, Henry Jiménez, Andrea Mishra, Suchismita Quiroz, Daniel A. Yépez, Mauricio |
| author_facet | Echeverría, Henry Jiménez, Andrea Mishra, Suchismita Quiroz, Daniel A. Yépez, Mauricio |
| contents | For a graph $G$, let $im(G)$ denote the maximum integer $t$ such that $G$ contains $K_t$ as an immersion. A recent paper of Collins, Heenehan, and McDonald (2023) studied the behaviour of this parameter under graph products, asking how large can $im(G\ast H)$ be in terms of $im(G)$ and $im(H)$, when $\ast$ is one of the four standard graph products. We consider a similar question for the parameter $toi(G)$ which denotes the maximum integer $t$ such that $G$ contains $K_t$ as a totally odd immersion. As an application, we obtain that no minimum counterexample to the immersion-analogue of the Odd Hadwiger Conjecture can be obtained from the Cartesian, direct (tensor), lexicographic or strong product of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Totally odd immersions of complete graphs in graph products Echeverría, Henry Jiménez, Andrea Mishra, Suchismita Quiroz, Daniel A. Yépez, Mauricio Combinatorics For a graph $G$, let $im(G)$ denote the maximum integer $t$ such that $G$ contains $K_t$ as an immersion. A recent paper of Collins, Heenehan, and McDonald (2023) studied the behaviour of this parameter under graph products, asking how large can $im(G\ast H)$ be in terms of $im(G)$ and $im(H)$, when $\ast$ is one of the four standard graph products. We consider a similar question for the parameter $toi(G)$ which denotes the maximum integer $t$ such that $G$ contains $K_t$ as a totally odd immersion. As an application, we obtain that no minimum counterexample to the immersion-analogue of the Odd Hadwiger Conjecture can be obtained from the Cartesian, direct (tensor), lexicographic or strong product of graphs. |
| title | Totally odd immersions of complete graphs in graph products |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.10227 |