On the size of two families of unlabeled bipartite graphs

Abstract

Let Bu(n,r) denote the set of unlabeled bipartite graphs whose edges connect a set of n vertices with a set of r vertices. In this paper, we provide exact formulas for |Bu(2,r)| and |Bu(3,r)| using Polya's Counting Theorem. Extending these results to n≥4 involves solving a set of complex recurrences and remains open. In particular, the number of recurrences that must be solved to compute |Bu(n,r)| is given by the number of partitions of n that is known to increase exponentially with n by Ramanujan-Hardy-Rademacher's asymptotic formula. © 2017 Kalasalingam University