One-factorizations of complete multipartite graphs with distance constraints

TL;DR

This paper explores the decomposition of complete multipartite graphs into subgraphs with minimum distance constraints, linking graph factorizations to optimal coding theory solutions.

Contribution

It introduces the study of one-factorizations with distance constraints in complete multipartite graphs and provides new decomposition results under specific conditions.

Findings

For n ≤ g, K_{n×g} decomposes into g^2 subgraphs with minimum distance three.

For even gn with n > g, K_{n×g} decomposes into g(n-1) one-factors with minimum distance three.

No such decomposition exists when gn is odd and n > g.

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).