On $4$-covers of cubic graphs with two adjacent odd circuits in a $2$-factor

TL;DR

This paper proves that certain cubic graphs with specific 2-factors can be covered by four perfect matchings, supporting the 7/5-Conjecture of Alon and Tarsi.

Contribution

It demonstrates that cubic graphs with a 2-factor of two odd circuits and three spokes in the complementary 1-factor can be covered by four perfect matchings.

Findings

Four perfect matchings cover the specified cubic graphs.

The result confirms the 7/5-Conjecture for these graphs.

Supports the broader conjecture in graph theory.

Abstract

Let $G$ be a cubic graph admitting a $2$-factor consisting of exactly two odd circuits, and let the complementary $1$-factor contain precisely three spokes (along with an arbitrary number of chords). We show that four perfect matchings can cover $G$. As a consequence, $G$ fulfils the 7/5-Conjecture of Alon and Tarsi.