Stability of graph pairs involving cycles

TL;DR

This work investigates when a graph pair $(Γ, C_n)$ is stable or nontrivially unstable, without requiring regularity of $Γ$. It develops a Σ-automorphism framework, introduces a compatibility notion via the auxiliary graph $Γ^*$ and the sets $L(u)$, and ties stability to the existence of non-diagonal $C_n$-automorphisms of $Γ$. The authors establish two main sufficient conditions for nontrivial instability when $n \neq 4$ and derive a comprehensive set of results linking even and odd cycle lengths to stability in the non-bipartite setting, including equivalences that involve $(Γ, K_2)$. They also introduce the concept of expected automorphisms $R(Γ, C_n)$ and show how the presence or absence of nontrivial automorphisms governs the stability of the direct product, with implications for symmetry analysis in networks and combinatorial graph theory.

Abstract

A graph pair $(Γ, Σ)$ is called stable if $\aut(Γ)\times\aut(Σ)$ is isomorphic to $\aut(Γ\timesΣ)$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. A graph is called $R$-thin if distinct vertices have different neighbourhoods. $Γ$ and $Σ$ are said to be coprime if there is no nontrivial graph $Δ$ such that $Γ\cong Γ_1 \times Δ$ and $Σ\cong Σ_1 \times Δ$ for some graphs $Γ_1$ and $Σ_1$. An unstable graph pair $(Γ, Σ)$ is called nontrivially unstable if $Γ$ and $Σ$ are $R$-thin connected coprime graphs and at least one of them is non-bipartite. This paper contributes to the study of the stability of graph pairs with a focus on the case when $Σ= C_n$ is a cycle. We give two sufficient conditions for $(Γ, C_n)$ to be nontrivially unstable, where $n \ne 4$ and $Γ$ is an $R$-thin connected graph. In the case when $Γ$ is an $R$-thin connected non-bipartite graph, we obtain the following results: (i) if $(Γ, K_2)$ is unstable, then $(Γ, C_{n})$ is unstable for every even integer $n \geq 4$; (ii) if an even integer $n \ge 6$ is compatible with $Γ$ in some sense, then $(Γ, C_{n})$ is nontrivially unstable if and only if $(Γ, K_2)$ is unstable; (iii) if there is an even integer $n \ge 6$ compatible with $Γ$ such that $(Γ, C_{n})$ is nontrivially unstable, then $(Γ, C_{m})$ is unstable for all even integers $m \ge 6$. We also prove that if $Γ$ is an $R$-thin connected graph and $n \ge 3$ is an odd integer compatible with $Γ$, then $(Γ, C_{n})$ is stable.

Figures (1)

• Figure 1: A graph $\Gamma$ (left) and its auxiliary graph $\Gamma^*$ (right).

Theorems & Definitions (29)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Example 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4