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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2509.26170 |
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| _version_ | 1866914589106503680 |
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| author | Wang, Xiaomeng Qin, Yan-Li Xia, Binzhou |
| author_facet | Wang, Xiaomeng Qin, Yan-Li Xia, Binzhou |
| contents | A pair of graphs $(Γ,Σ)$ is called unstable if their direct product $Γ\timesΣ$ admits automorphisms not from $\mathrm{Aut}(Γ)\times\mathrm{Aut}(Σ)$, and such automorphisms are said to be unexpected. The stability of a graph $Γ$ refers to that of $(Γ,K_2)$. While the stability of individual graphs has been relatively well studied, much less is known for graph pairs. In this paper, we propose a conjecture that provides the best possible reduction of the stability of a graph pair to the stability of a single graph. We prove one direction of this conjecture and establish partial results for the converse. This enables the determination of the stability of a broad class of graph pairs, with complete results when one factor is a cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The existence of unexpected automorphisms in direct product graphs Wang, Xiaomeng Qin, Yan-Li Xia, Binzhou Combinatorics A pair of graphs $(Γ,Σ)$ is called unstable if their direct product $Γ\timesΣ$ admits automorphisms not from $\mathrm{Aut}(Γ)\times\mathrm{Aut}(Σ)$, and such automorphisms are said to be unexpected. The stability of a graph $Γ$ refers to that of $(Γ,K_2)$. While the stability of individual graphs has been relatively well studied, much less is known for graph pairs. In this paper, we propose a conjecture that provides the best possible reduction of the stability of a graph pair to the stability of a single graph. We prove one direction of this conjecture and establish partial results for the converse. This enables the determination of the stability of a broad class of graph pairs, with complete results when one factor is a cycle. |
| title | The existence of unexpected automorphisms in direct product graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.26170 |