06118538

j

06118538 Gosselin, Shonda Self-complementary non-uniform hypergraphs. Graphs Comb. 28, No. 5, 615-635 (2012). 2012 Springer-Verlag, Tokyo EN self-complementary hypergraph regular hypergraph transitive hypergraph large set of $t$-designs

doi:10.1007/s00373-011-1070-x

Summary: Let $V$ be a finite set. For a nonempty subset $K$ of positive integers, a $K$-hypergraph on $V$ is a hypergraph with vertex set $V$ and edge set ${E=\bigcup_{k\in K}E_k}$, where $E _{k }$ is a nonempty set of $k$-subsets of $V$. We define the complement of a $K$-hypergraph $(V, E)$ to be the $K$-hypergraph on $V$ whose edge set consists of the subsets of $V$ with cardinality in $K$ which do not lie in $E$. A $K$-hypergraph is called self-complementary if it is isomorphic to its complement. The two extreme classes of self-complementary $K$-hypergraphs have been studied previously. When $|K| = 1$ these are the self-complementary uniform hypergraphs, and when $|K| = |V| - 1$, these are the so called `self-complementary hypergraphs' studied by Zwonek. In this paper we determine necessary conditions on the order of self-complementary $K$-hypergraphs, and on the order of regular or vertex-transitive self-complementary $K$-hypergraphs, for various sets of positive integers $K$. We also present several constructions for $K$-hypergraphs to show that these necessary conditions are sufficient for certain sets $K$. In the language of design theory, the $t$-subset-regular self-complementary $K$-hypergraphs correspond to large sets of two isomorphic $t$-wise balanced designs, or $t$-partitions, in which the block sizes lie in the set $K$. Hence the results of this paper imply results in design theory.