The maximum number of perfect matchings of semi-regular graphs

TL;DR

This paper proves that certain semi-regular graphs with degrees close to half the order contain a maximum number of disjoint perfect matchings, extending and improving previous results in graph factorization.

Contribution

It establishes the maximum number of disjoint perfect matchings in semi-regular graphs with degrees in a specific range, generalizing and strengthening prior theorems.

Findings

Every $ extstyleigracevert D_n, D_n+1 igracevert$-graph of order $n$ contains $ extstyleigracevert n/4 igracevert$ disjoint perfect matchings.

The result is sharp; there exist graphs with exactly $ extstyleigracevert n/4 igracevert$ disjoint perfect matchings.

Extends and improves upon previous theorems by Csaba et al., Zhang and Zhu, and Hou.

Abstract

Let $n\ge 34$ be an even integer, and $D_n=2\lceil n/4 \rceil-1$. In this paper, we prove that every $\{D_n,\,D_n+1\}$-graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings. This result is sharp in the sense that (i) there exists a $\{D_n,\,D_n+1\}$-graph containing exactly $\lceil n/4 \rceil$ disjoint perfect matchings, and that (ii) there exists a $\{D_n-1,\,D_n\}$-graph without perfect matchings for each $n$. As a consequence, for any integer $D\ge D_n$, every $\{D,\,D+1\}$-graph of order $n$ contains $\lceil (D+1)/2 \rceil$ disjoint perfect matchings. This extends Csaba et~al.'s breathe-taking result that every $D$-regular graph of sufficiently large order is $1$-factorizable, generalizes Zhang and Zhu's result that every $D_n$-regular graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect matchings, and improves Hou's result that for all $k\ge n/2$, every $\{k,\,k+1\}$-graph of order $n$ contains $(\lfloor n/3\rfloor+1+k-n/2)$ disjoint perfect matchings.