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The existence of bipartite almost self-complementary 3-uniform hypergraphs

Kamble L. N., Deshpande C. M., Athawale B. P.

Opuscula Mathematica

2023

Vol. 43, no. 5

663--673

An almost self-complementary 3-uniform hypergraph on n vertices exists if and only if n is congruent to 3 modulo 4. A hypergraph H with vertex set V and edge set E is called bipartite if V can be partitioned into two subsets V1 and V2 such that e ∩ V1 ̸= ∅ and e ∩ V2̸= ∅ for any e ∈ E. A bipartite self-complementary 3-uniform hypergraph H with partition (V1, V2) of the vertex set V such that |V1| = m and |V2| = n exists if and only if either (i) m = n or (ii) m ̸= n and either m or n is congruent to 0 modulo 4 or (iii) m ̸= n and both m and n are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph H with partition (V1, V2) of a vertex set V such that |V1| = m and |V2| = n and find the conditions on m and n for a bipartite 3-uniform hypergraph H to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs.