Odd 2-factored snarks

TL;DR

This paper introduces a method for constructing odd 2-factored snarks, presents two new families that counter previous conjectures, and offers partial characterizations and new conjectures about these complex graphs.

Contribution

It provides a novel construction method for odd 2-factored snarks, including two counterexample families, and advances understanding through partial characterization and new conjectures.

Findings

Constructed two families of odd 2-factored snarks that disprove existing conjectures.

Developed a method for constructing odd 2-factored snarks.

Provided partial characterization of cyclically 4-edge connected odd 2-factored snarks.

Abstract

A {\em snark} is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph $G$ is {\em odd 2-factored} if for each 2-factor F of G each cycle of F is odd. In this paper, we present a method for constructing odd 2--factored snarks. In particular, we construct two families of odd 2-factored snarks that disprove a conjecture by some of the authors. Moreover, we approach the problem of characterizing odd 2-factored snarks furnishing a partial characterization of cyclically 4-edge connected odd 2-factored snarks. Finally, we pose a new conjecture regarding odd 2-factored snarks.