Hypohamiltonian planar cubic graphs with girth five

TL;DR

This paper introduces the first known hypohamiltonian planar cubic graphs with girth five, constructed via computer search, expanding the understanding of such graphs beyond previous examples with 4-cycles.

Contribution

It presents the first examples of hypohamiltonian planar cubic graphs with girth five and cyclic connectivity five, including the smallest such graphs with 76 vertices.

Findings

First hypohamiltonian planar cubic graphs with girth five

Three smallest graphs with 76 vertices identified

Computer search used to find these new graphs

Abstract

A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Many hypohamiltonian planar cubic graphs have been found, starting with constructions of Thomassen in 1981. However, all the examples found until now had 4-cycles. In this note we present the first examples of hypohamiltonian planar cubic graphs with cyclic connectivity five, and thus girth five. We show by computer search that the smallest members of this class are three graphs with 76 vertices.