The existence of unexpected automorphisms in direct product graphs
Xiaomeng Wang, Yan-Li Qin, Binzhou Xia
TL;DR
This work advances the stability theory of direct product graphs by formulating and testing a comprehensive conjecture: for nontrivial graph pairs with a bipartite second factor, stability of the pair should be equivalent to stability of the first factor. The paper develops two key tools—two-fold-automorphisms (TF-automorphisms) and partitions in Cartesian products—to translate automorphism issues into mixer phenomena and partition arguments, enabling structural decompositions and transfer of stability properties. It proves one direction of the conjecture and provides substantial partial results for the converse, including a thorough treatment of cycles: for odd cycles no nontrivially unstable pairs occur, while for even cycles a precise coprimality condition characterizes instability. Together with a gap-fixing analysis and explicit constructions, the results yield broad, applicable criteria to determine the stability of many graph pairs, particularly when one factor is a cycle, and pave the way toward a full resolution of the conjecture.
Abstract
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.