One-factorizations of complete multipartite graphs with distance constraints
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911454154719232 |
|---|---|
| author | Tan, Yuli Zhou, Junling Etzion, Tuvi |
| author_facet | Tan, Yuli Zhou, Junling Etzion, Tuvi |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16319 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | One-factorizations of complete multipartite graphs with distance constraints Tan, Yuli Zhou, Junling Etzion, Tuvi Combinatorics 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). |
| title | One-factorizations of complete multipartite graphs with distance constraints |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.16319 |