On The Number Of Labeled Bipartite Graphs

Let $I$ and $O$ denote two sets of vertices, where $I\cap O=\mathrm{\varnothing }$, $\mathrm{\mid }I\mathrm{\mid }=n$, $\mathrm{\mid }O\mathrm{\mid }=r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. Recently, it was established in\cite{atmacaoruc2018} that the following two-sided equality holds, \begin{equation} \label{mainResult} \displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!}, \le, |B_u(n,r)|, \le, 2\frac{\binom{r+2^{n}-1}{r}}{n!},, n < r.\nonumber \end{equation} and exact formulas were provided in~\cite{atmaca2017size} for $\mathrm{\mid }{B}_u(2,r)\mathrm{\mid }$ and $\mathrm{\mid }{B}_u(3,r)\mathrm{\mid }\mathrm{.}$ In this paper, these results are extended to various families of labeled bipartite graphs.