One-factorizations of complete multipartite graphs with distance constraints

TL;DR

It is proved that the complete multipartite graph K_{n\times g} can be decomposed into $g(n-1)$ one-factors with minimum distance three, leaving a small gap of $n$ (in terms of $g$) to be resolved.

Abstract

The present paper considers multipartite graphs from the perspective of design theory and coding theory. A one-factor $F$ of the complete multipartite graph $K_{n\times g}$ (with $n$ parts of size $g$) gives rise to a $(g+1)$-ary code ${\cal C}$ of length $n$ and constant weight two. Furthermore, if the one-factor $F$ meets a certain constraint, then ${\cal C}$ becomes an optimal code with minimum distance three. We initiate the study of one-factorizations of complete multipartite graphs subject to distance constraints. The problem of decomposing $K_{n\times g}$ into the largest subgraphs with minimum distance three is investigated. It is proved that, for $n\le g$, the complete multipartite graph $K_{n\times g}$ can be decomposed into $g^2$ copies of the largest subgraphs with minimum distance three. For even $gn$ with $n>g$, it is proved that the complete multipartite graph $K_{n\times g}$ can be decomposed into $g(n-1)$ one-factors with minimum distance three, leaving a small gap of $n$ (in terms of $g$) to be resolved (If $gn$ is odd when $n>g$, no such decomposition of $K_{n\times g}$ exists).

Theorems & Definitions (24)

• Lemma 1.1: bound_d2
• Lemma 1.2: Chee-2007
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Example 2.2
• Theorem 2.3
• Example 2.4
• Theorem 2.5
• Example 2.6