M\'acajov\'a and \v{S}koviera Conjecture on Cubic Graphs

TL;DR

This paper proves the Mácajová and Škoviera conjecture for small cubic graphs and establishes a stronger result for traceable graphs, advancing understanding of perfect matchings in cubic graphs.

Contribution

It confirms the conjecture for graphs with few vertices and provides an improved result for traceable cubic graphs, contributing to the theory of perfect matchings.

Findings

Confirmed the conjecture for small cubic graphs

Established a stronger result for traceable graphs

Enhanced understanding of perfect matchings in cubic graphs

Abstract

A conjecture of M\'a\u{c}ajov\'a and \u{S}koviera asserts that every bridgeless cubic graph has two perfect matchings whose intersection does not contain any odd edge cut. We prove this conjecture for graphs with few vertices and we give a stronger result for traceable graphs.