The Minimality of the Georges-Kelmans Graph

TL;DR

This paper proves the minimality of the Georges-Kelmans graph as the smallest non-hamiltonian 3-connected bipartite cubic graph and constructs an infinite family of such graphs with higher connectivity.

Contribution

It establishes the minimal size of non-hamiltonian bipartite cubic graphs and introduces a new infinite family with cyclically 5-connected properties.

Findings

The Georges-Kelmans graph is the smallest non-hamiltonian 3-connected bipartite cubic graph.

An infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs is constructed.

Barnette's conjecture holds for graphs up to at least 90 vertices.

Abstract

In 1971, Tutte wrote in an article that "it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian". Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts -- even if they have girth 6 -- it is natural to ask whether this is a necessary prerequisite. In this article we answer this question in the negative and give a construction of an infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs. In 1969, Barnette gave a weaker version of the conjecture stating that 3-connected planar bipartite cubic graphs are hamiltonian. We show that Barnette's conjecture is true up to at least 90 vertices. We also report that a search of small non-hamiltonian 3-connected bipartite cubic graphs did not find any with genus less than 4.