A Ten-Vertex Counterexample to a Conjecture on Unstable Graphs

Prateek R. Srivastava

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}.

Disclosure

“nstructs the graph from (2), checks all ten neighbourhood identities, checks the induced automorphism of the canonical double cover, and enumerates every simple cycle. The example, verifier, and drafting were developed with assistance from OpenAI Codex. The proof above is self-contained, and the author takes responsibility for all claims. References [1] J. Lauri, R. Mizzi, and R. Scapellato, Unstable graphs: a fresh outlook via TF-automorphisms, Ars Math. Contemp. 8 (2015), 1”
PDF page 2

Classification: Drafting limited passages
Multiplier: 5
Verified: 2026-08-23T05:16:38.511619Z

Structural counts

Pages 2 pdf

Theorems 1 source

Lemmas 0 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 8 source

Bibliography entries 2 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.