Lifting and Folding: A Framework for Unstable Graphs and TF-Cousins

Abstract

A graph $G$ is \emph{unstable} if its canonical double cover CDC$(G)$ has more automorphisms than Aut$(G)\times \mathbb{Z}_2$. A related problem asks when two non-isomorphic graphs share the same CDC. We unify both via \emph{lifting} and \emph{guided folding}, showing that they are governed by conjugacy classes of strongly switching involutions in Aut(\CDC$(G)$). Using \emph{two-fold isomorphisms} (TF-isomorphisms), lifting $(α,β):G\to H$ produces a digraph isomorphic to the alternating double cover of $G$, while folding yields a graph TF-isomorphic to $G$. If this graph is non-isomorphic to $G$, the pair forms TF-cousins; otherwise $(α,β)$ is a non-trivial TF-automorphism and $G$ is unstable. Distinct conjugacy classes of switching involutions in Aut$(CDC(G))$ produce non-isomorphic graphs with a common CDC, recovering a theorem of Pacco and Scapellato. The framework generates TF-cousin pairs and unstable graphs of arbitrary order from $(C_k\cup C_k,\, C_{2k})$. We introduce the \emph{claw graph} family CG$(n)$ and show that CG$(n)$ and CG'$(n)$ are TF-cousins iff $n$ is odd. For $n=1$, this yields the Petersen graph and a cubic companion on $10$ vertices, both with the Desargues graph as CDC. For odd $n\geq 3$, we obtain new non-isomorphic cubic graphs sharing a CDC. We conjecture that every TF-cousin pair and unstable graph contains cycles $C_k$ and $C_{2k}$ for some odd $k$, verified for all connected graphs on at most $9$ vertices.

Figures (10)

• Figure 1: A non-trivial TF-isomorphism from $G$ to $H$, with $\alpha=(2\ 5)$ and $\beta=(1\ 4)(3\ 6)$.
• Figure 2: The smallest known unstable asymmetric graph cockroachpaper. With $\gamma=(1\ 2\ 3)(4\ 5\ 6)(7\ 8\ 9)(10\ 11\ 12)$, the pair $(\gamma,\gamma^{-1})$ is a non-trivial TF-automorphism.
• Figure 3: The graph $G\cong C_3$ (left) and its lift $\vec{G}_{\alpha,\beta}\cong\hbox{\rm{ADC}}(G)$ (right).
• Figure 4: A TF-isomorphism from $C_3\cup C_3$ to $C_6$.
• Figure 5: The lift $\vec{G_0}_{\alpha,\beta}$ corresponding to Figure \ref{['fig:genesis01']}.

Theorems & Definitions (26)

• Theorem 2.1: Ars
• Theorem 3.1: lms1
• Theorem 3.2: dissertation
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Example 4.1
• Lemma 4.3
• proof