Canonical double covers of circulants

TL;DR

This paper proves Qin et al.'s conjecture that all circulant graphs of odd order are stable, meaning their automorphism groups behave predictably with their canonical double covers.

Contribution

The paper confirms that no nontrivially unstable circulant graphs of odd order exist, settling a conjecture in the field.

Findings

All circulant graphs of odd order are stable.

Nontrivially unstable circulant graphs of odd order do not exist.

The automorphism group behavior of circulants aligns with stability conjecture.

Abstract

The canonical double cover $B(X)$ of a graph $X$ is the direct product of $X$ and $K_2$. If $Aut(B(X)) \cong Aut(X) \times \mathbb{Z}_2$ then $X$ is called stable; otherwise $X$ is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. Circulant is a Cayley graph on a cyclic group. Qin et al. conjectured in [J. Combin. Theory Ser. B 136 (2019), 154-169] that there are no nontrivialy unstable circulants of odd order. In this paper we prove this conjecture.