Some Infinite Families of t-Regular Self-complementary k-Hypergraphs

Abstract

A t-regular self-complementary k-hypergraph, denoted by $\hbox {SRHG}(t,k,v)$, is a k-hypergraph with a v-set V as vertex set and an edge set E, such that every t-subset of V lies in the same number of edges and there is a permutation $\sigma \in S_v$ with the property that $e\in E$ if and only if $\sigma (e)\notin E$. It is clear that a set of trivial necessary conditions for the existence of an $\hbox {SRHG}(t,k,v)$ is that ${v-i\atopwithdelims ()k-i}$ is an even integer for all $i=0,1,...,t$. In this paper, we extend the method of partitionable sets for constructing large sets of t-designs to obtain new $\hbox {SRHG}(t,k,v)$. In particular, we present $\hbox {SRHG}(2,3,10)$, $\hbox {SRHG}(2,4,10)$, $\hbox {SRHG}(2,4,11)$ and $\hbox {SRHG}(2,5,10)$. Also we show that the trivial necessary conditions for the existence of $\hbox {SRHG}(2,k,v)$ with $k\le 7$ are sufficient.