Counterexamples to Clique Immersion Conjecture for Direct Products
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911726036844544 |
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| author | Wu, Chuanshu Deng, Zijian |
| author_facet | Wu, Chuanshu Deng, Zijian |
| contents | Let \(G\) and \(H\) be graphs, and let \(G\times H\) denote their direct product. For a graph \(G\), let \(\operatorname{im}(G)\) be the largest integer \(t\) such that \(G\) contains a \(K_t\)-immersion. Collins, Heenehan, and McDonald conjectured that if \(\operatorname{im}(G)=t\) and \(\operatorname{im}(H)=r\), then \[\operatorname{im}(G\times H)\ge (t-1)(r-1)+1.\] We disprove this conjecture by constructing an infinite family of connected bipartite counterexamples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28518 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counterexamples to Clique Immersion Conjecture for Direct Products Wu, Chuanshu Deng, Zijian Combinatorics Let \(G\) and \(H\) be graphs, and let \(G\times H\) denote their direct product. For a graph \(G\), let \(\operatorname{im}(G)\) be the largest integer \(t\) such that \(G\) contains a \(K_t\)-immersion. Collins, Heenehan, and McDonald conjectured that if \(\operatorname{im}(G)=t\) and \(\operatorname{im}(H)=r\), then \[\operatorname{im}(G\times H)\ge (t-1)(r-1)+1.\] We disprove this conjecture by constructing an infinite family of connected bipartite counterexamples. |
| title | Counterexamples to Clique Immersion Conjecture for Direct Products |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.28518 |