Perfect partition of some regular bipartite graphs

TL;DR

This paper investigates perfect partitions in certain regular bipartite graphs, providing formulas for perfect matchings and demonstrating the existence of perfect partitions in specific cases.

Contribution

It introduces a formula for counting perfect matchings in $L_{rm, r}$ graphs and proves the existence of perfect partitions for $L_{6,1}$ and $L_{8,2}$.

Findings

Derived a formula for perfect matchings in $L_{rm, r}$.

Proved $L_{6,1}$ has a perfect partition.

Proved $L_{8,2}$ has a perfect partition.

Abstract

A graph has a perfect partition if all its perfect matchings can be partitioned so that each part is a 1-factorization of the graph. Let $L_{rm, r}=K_{rm,rm}-mK_{r,r}$. We first give a formula to count the number of perfect matchings of $L_{rm, r}$, then show that $L_{6,1}$ and $L_{8,2}$ have perfect partitions.