The Existence of Regular and Quasi-regular Bipartite Self-complementary 3-uniform Hypergraphs

Abstract

A hypergraph $H$ with vertex set $V$ and edge set $E$ is called bipartite if $V$ can be partitioned into two subsets $V_1$ and $V_2$ such that $e\cap V_1\ne \varphi$ and $e\cap V_2\ne \varphi$ for any $e\in E$. A bipartite self-complementary 3-uniform hypergraph $H$ with partition $(V_1,V_2)$ of a vertex set $V$ such that $|V_1|=m$ and $|V_2|=n$ exists if and only if either (i) $m=n$ or (ii) $m\ne n$ and either $m$ or $n$ is congruent to 0 modulo 4 or (iii) $m\ne n$ and both $m$ and $n$ are congruent to 1 or 2 modulo 4.

In this paper we prove that, there exists a regular bipartite self-complementary 3-uniform hypergraph $H(V_1,V_2)$ with $|V_1|=m,|V_2|=n,m+n>3$ if and only if $m=n$ and $n$ is congruent to 0 or 1 modulo 4. Further we prove that, there exists a quasi-regular bipartite self-complementary 3-uniform hypergraph $H(V_1,V_2)$ with $|V_1|=m,|V_2|=n,m+n>3$ if and only if either $m=3,n=4$ or $m=n$ and $n$ is congruent to 2 or 3 modulo 4.