An Asymptotic Version of the Multigraph 1-Factorization Conjecture

TL;DR

This paper proves that large regular multigraphs with bounded multiplicity and sufficiently high degree are 1-factorizable, extending previous results to a broader class of graphs with an asymptotic approach.

Contribution

It provides a self-contained proof establishing an asymptotic version of the multigraph 1-factorization conjecture for all positive integers r and epsilon.

Findings

Large regular multigraphs are 1-factorizable under specified conditions

Extends previous results to multigraphs with bounded multiplicity

Provides a new asymptotic proof approach

Abstract

We give a self-contained proof that for all positive integers $r$ and all $\epsilon > 0$, there is an integer $N = N(r, \epsilon)$ such that for all $n \ge N$ any regular multigraph of order $2n$ with multiplicity at most $r$ and degree at least $(1+\epsilon)rn$ is 1-factorizable. This generalizes results of Perkovi{\'c} and Reed, and Plantholt and Tipnis.