Orthogonally Resolvable Matching Designs

TL;DR

This paper characterizes the existence conditions for Orthogonally Resolvable Matching Designs, which partition complete graph edges into matchings with specific resolution properties, providing a complete existence criterion.

Contribution

It establishes necessary and sufficient conditions for the existence of OMD$(n,k)$, filling a gap in combinatorial design theory.

Findings

OMD$(n,k)$ exists if and only if $n ot\equiv 0 mod 2k$, except for specific small cases.

Provides a complete characterization of the existence of these designs.

Clarifies the structure and constraints of orthogonally resolvable matchings.

Abstract

An Orthogonally resolvable Matching Design OMD$(n, k)$ is a partition of the edges the complete graph $K_n$ into matchings of size $k$, called blocks, such that the blocks can be resolved in two different ways. Such a design can be represented as a square array whose cells are either empty or contain a matching of size $k$, where every vertex appears exactly once in each row and column. In this paper we show that an OMD$(n.k)$ exists if and only if $n \equiv 0 \pmod{2k}$ except when $k=1$ and $n = 4$ or $6$.