4-Hyperclique Decompositions of 3-Hyperdistance Multihypergraphs of Multipartite Graphs

Chutima Saengchampa, Chariya Uiyyasathian

Abstract

This paper investigates hyperclique decompositions of a hypergraph transformation, namely the hyperdistance multihypergraph defined as follows: Given an integer $k\geq 3$ and any hypergraph $H$, the \textit{ $k$-hyperdistance multihypergraph} of $H$, denoted by $D^{(k)}(H)$, is the $k$-uniform hypergraph which has  the same vertex set as $H$ and for any $k$-subset $\{v_1, $$v_2, $$\ldots v_k\}$ of $V(H)$, $D^{(k)}(H)$ has exactly $\sum_{i \neq j}^{} d_H(v_i, v_j)$ copies of hyperedges $\{v_1, v_2, \ldots, v_k\}$ where $d_H(v_i, v_j)$ is the distance between vertices $v_i$ and $v_j$ in $H$. We study $4$-hyperclique decompositions of $D^{(3)}(H)$ where $H$ is a complete multipartite graph. Our construction technique includes classic designs such as Latin cubes and factorizations of graphs.