Planar Hypohamiltonian Graphs on 40 Vertices

TL;DR

This paper improves the known minimum size of planar hypohamiltonian graphs to 40 vertices, introduces new constructions, and explores related hypotraceable graphs, advancing understanding of these special graph classes.

Contribution

The authors discover 25 new planar hypohamiltonian graphs of order 40, reducing the known minimum size and demonstrating existence for all larger orders.

Findings

Existence of planar hypohamiltonian graphs with 40 vertices.

Existence of planar hypotraceable graphs for all orders ≥ 154.

Smallest planar hypohamiltonian graph with girth 5 has 45 vertices.

Abstract

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest hypohamiltonian planar graph of girth 5 has 45 vertices.