2024/02/12 by Atmaca, Abdullah, Oruc, A. Yavuz
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2402.08053
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\citeatmacaoruc2018 that the following two-sided equality holds, \frac\binomr+2n-1rn! ≤ |Bu(n,r)| ≤ 2\frac\binomr+2n-1rn!, n lt; r.and exact formulas were provided in~\citeatmaca2017size for |Bu(2,r)| and |Bu(3,r)|. In this paper, these results are extended to various families of labeled bipartite graphs.