On disjoint matchings in cubic graphs

TL;DR

This paper investigates bounds on the maximum edges covered by disjoint matchings in cubic graphs, establishing new lower bounds for two and three matchings and a relation between these bounds.

Contribution

It provides improved lower bounds for the maximum edges covered by two and three disjoint matchings in cubic graphs and relates these bounds mathematically.

Findings

4/5 of vertices can be covered by two matchings

7/6 of vertices can be covered by three matchings

A new inequality relating 4/5 and 7/6 bounds

Abstract

For $i=2,3$ and a cubic graph $G$ let $\nu_{i}(G)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that $\nu_{2}(G)\geq {4/5}| V(G)| $ and $\nu_{3}(G)\geq {7/6}| V(G)| $. Moreover, it turns out that $\nu_{2}(G)\leq \frac{|V(G)|+2\nu_{3}(G)}{4}$.