Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

TL;DR

This paper characterizes irreducible pseudo 2--factor isomorphic cubic bipartite graphs, showing only the Heawood and Pappus graphs fit this category, and offers partial support for a broader conjecture.

Contribution

It provides a complete characterization of irreducible pseudo 2--factor isomorphic cubic bipartite graphs, identifying only the Heawood and Pappus graphs as such.

Findings

Only the Heawood and Pappus graphs are irreducible pseudo 2--factor isomorphic Levi graphs.

The characterization supports a conjecture about essentially 4--edge-connected cubic bipartite graphs.

Partial proof of the conjecture that the only such graphs are K_{3,3}, Heawood, and Pappus graphs.

Abstract

A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors have the same parity of number of circuits. In \cite{ADJLS} we proved that the only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite graph of girth 4 is $K_{3,3}$, and conjectured \cite[Conjecture 3.6]{ADJLS} that the only essentially 4--edge-connected cubic bipartite graphs are $K_{3,3}$, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations $n_3$ %{\bf decide notation and how to use it in the rest of the paper} due to Martinetti (1886) in which all symmetric configurations $n_3$ can be obtained from an infinite set of so called {\em irreducible} configurations \cite{VM}. The list of irreducible configurations has been completed by Boben \cite{B} in terms of their {\em irreducible Levi graphs}. In this paper we characterize irreducible pseudo 2--factor isomorphic cubic bipartite graphs proving that the only pseudo 2--factor isomorphic irreducible Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained characterization allows us to partially prove the above Conjecture.