Edge-connectivity and pairwise disjoint perfect matchings in regular graphs

TL;DR

This paper investigates the maximum number of pairwise disjoint perfect matchings in regular graphs with given edge connectivity, providing new bounds and extending known results in graph theory.

Contribution

It establishes new upper bounds for the number of disjoint perfect matchings in regular graphs with specific connectivity and degree conditions.

Findings

Proves that m(2l,r) ≤ 3l - 6 for l ≥ 3 and r ≥ 2l.

Extends known bounds for pairwise disjoint perfect matchings in regular graphs.

Provides new insights into the structure of highly connected regular graphs.

Abstract

For $0 \leq t \leq r$ let $m(t,r)$ be the maximum number $s$ such that every $t$-edge-connected $r$-graph has $s$ pairwise disjoint perfect matchings. There are only a few values of $m(t,r)$ known, for instance $m(3,3)=m(4,r)=1$, and $m(t,r) \leq r-2$ for all $t \not = 5$, and $m(t,r) \leq r-3$ if $r$ is even. We prove that $m(2l,r) \leq 3l - 6$ for every $l \geq 3$ and $r \geq 2 l$.