On The Number Of Labeled Bipartite Graphs

Let I and O denote two sets of vertices, where I∩ O =∅, $|I| = n$, $|O| = r$, and Bᵤ(n,r) denote the set of unlabeled graphs whose edges connect vertices in I and O. Recently, it was established in{atmacaoruc2018} that the following two-sided equality holds, {equation} {mainResult} {{r+2ⁿ-1}{r}}{n!}\, ≤\, |B_u(n,r)|\, ≤\, 2{{r+2ⁿ-1}{r}}{n!},\, n < r. {equation} and exact formulas were provided in~{atmaca2017size} for |Bᵤ(2,r)| and |Bᵤ(3,r)|. In this paper, these results are extended to various families of labeled bipartite graphs.