Abstract:A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The stability of a graph $\Gamma$ refers to that of $(\Gamma,K_2)$. 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.
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2509.26170 [math.CO] |
| (or arXiv:2509.26170v2 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2509.26170 arXiv-issued DOI via DataCite |
From: XiaoMeng Wang [view email]
[v1]
Tue, 30 Sep 2025 12:25:20 UTC (19 KB)
[v2]
Fri, 22 May 2026 02:02:35 UTC (19 KB)
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