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

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 $ V_1 $ and $ V_2 $ such that $ e ∩ V_1 ≠ ∅ $ and $e ∩ V_2 ≠ ∅ $ for any $ e ∈ E $. A bipartite self-complementary 3-uniform hypergraph $ H $ with partition $ (V_1, V_2) $ of the vertex set $ V $ such that $ |V_1| = m $ and $ |V_2| = 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 $ (V_1, V_2) $ of a vertex set $ V $ such that $ |V_1| = m $ and $ |V_2| = 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.