A Ten-Vertex Counterexample to a Conjecture on Unstable Graphs

Abstract

Mizzi conjectured that every nontrivially unstable graph contains cycles C_k and C_{2k} for some odd k. We give a connected, nonbipartite, vertex-determining counterexample on ten vertices. Its instability is certified by an explicit nontrivial two-fold automorphism, and its complete set of simple-cycle lengths is {5,5,6}.