Edge-connectivity and pairwise disjoint perfect matchings in regular graphs

TL;DR

It is proved that $m(2l,r) \leq 3l - 6$ for every $l \geq 3$ and $r £2 l$.

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 $r \geq 2 l$.

Figures (7)

• Figure 1: The Petersen graph $P$, and its perfect matchings $M_0$ and $M_i$.
• Figure 2: The operation of Definition \ref{['Definition-t-splicing']}.
• Figure 3: The graph $G=P+2M_0+M_1+M_2+M_3$ (left) and the graph $G'$ (right) constructed from $G$ in the proof of Lemma \ref{['lem:construction']}. The edges of $M$ and $M'$ respectively are drawn in bold red lines.
• Figure 4: The graph $Q_1$ (solid lines) and the edge sets $E_1^1$, $E_1^2$ (dashed lines). The bold red edges are used to construct $M^6$ in the proof of Theorem \ref{['theo:main_result']}.
• Figure 5: The graph $G_1$.

Theorems & Definitions (18)

• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof