Counterexamples to a conjecture of Hoa on maximal non-Hamiltonian graphs

Xingzhi Zhan

Counterexamples to a conjecture of Hoa on maximal non-Hamiltonian graphs

Combinatorics 2026-08-02 v1

Authors: Xingzhi Zhan

Abstract

A graph G is said to be maximal non-Hamiltonian if G is non-Hamiltonian, but $G+e$ is Hamiltonian for every nonedge $e$ of $G.$ In 1994, Vu Dinh Hoa conjectured that if $C$ is a longest cycle of a maximal non-Hamiltonian graph $G,$ then $G-V(C)$ is a complete graph. We disprove this conjecture by constructing a counterexample of every order $n\ge 56.$

Keywords

maximal non-Hamiltonian graphs longest cycle counterexample graph theory Hamiltonian graphs

Cite

@article{arxiv.2608.00957,
  title  = {Counterexamples to a conjecture of Hoa on maximal non-Hamiltonian graphs},
  author = {Xingzhi Zhan},
  journal= {arXiv preprint arXiv:2608.00957},
  year   = {2026}
}

Comments

8 pages, 3 figures