Formula omitted]-partite self-complementary and almost self-complementary [formula omitted]-uniform hypergraphs

Abstract

A hypergraph H is said to be r-partite r-uniform if its vertex set V can be partitioned into non-empty sets V1,V2,...,Vr so that every edge in the edge set E(H), consists of precisely one vertex from each set Vi, i=1,2,…,r. It is denoted as Hr(V1,V2,…,Vr) or H(n1,n2,…,nr)r if |Vi|=ni for i=1,2,…r. In this paper we define r-partite self-complementary and almost self-complementary r-uniform hypergraph. We prove that, there exists an r-partite self-complementary r-uniform hypergraph Hr(V1,V2,…,Vr) where |Vi|=ni for i=1,2,…,r if and only if at least one of n1,n2,…,nr is even. And we prove that, there exists an r-pasc Hr(V1,V2,…,Vr) where |Vi|=ni for i=1,2,…,r if and only if n1,n2,…,nr are odd. Further, we analyze the cycle structure of complementing permutations of r-partite self-complementary r-uniform hypergraphs and r-partite almost self-complementary r-uniform hypergraphs.