Matching Theory and Barnette's Conjecture

TL;DR

This paper translates Barnette's Conjecture into a matching-theoretic framework, relaxing planarity to Pfaffian properties, and reduces the problem to cubic, 3-connected, bipartite braces, with practical tools for graph generation and verification.

Contribution

It introduces a matching-theoretic perspective on Barnette's Conjecture, reducing it to cubic, 3-connected, bipartite braces, and provides practical methods for graph generation and verification.

Findings

Barnette's Conjecture can be reformulated in matching-theoretic terms.

The conjecture reduces to cubic, 3-connected, Pfaffian, bipartite graphs.

Practical tools are developed for checking if generated graphs are braces.

Abstract

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduction to braces for cubic, 3-connected, bipartite graphs regarding four stronger versions of Hamiltonicity. Note that in these cases we do not need planarity. As a practical application of these results, we provide some supplements to a generation procedure for cubic, 3-connected, planar, bipartite graphs discovered by Holton et al. [Hamiltonian Cycles in Cubic 3-Connected Bipartite Planar Graphs, JCTB, 1985]. These allow us to check whether a graph we generated is a brace.