Improved bounds for hypohamiltonian graphs

TL;DR

This paper develops an algorithm to generate hypohamiltonian graphs, establishing the exact counts for orders 18 and 19, and determining minimal sizes for graphs with specific properties such as girth, planarity, and cubic structure.

Contribution

It introduces a new algorithm for generating hypohamiltonian graphs and provides exact counts for certain orders, extending previous classifications and establishing new lower bounds for various graph properties.

Findings

Exactly 14 hypohamiltonian graphs of order 18

Exactly 34 hypohamiltonian graphs of order 19

Smallest planar hypohamiltonian graph has at least 23 vertices

Abstract

A graph $G$ is hypohamiltonian if $G$ is non-hamiltonian and $G - v$ is hamiltonian for every $v \in V(G)$. In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 graphs of order 19. We also extend their results in the cubic case. Furthermore, we show that (i) the smallest graph of girth 6 has order 25, (ii) the smallest planar graph has order at least 23, (iii) the smallest cubic planar graph has order at least 54, and (iv) the smallest cubic planar graph of girth 5 with non-trivial automorphism group has order 78.