Asymptotics for the number of bipartite graphs with fixed surplus
Abstract
In a recent work on the bipartite Erdos-R\'enyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in order to provide a probabilistic formula for the number of bipartite graphs with fixed surplus. Using this, we obtain asymptotics as the number of vertices in each class tend to infinity.
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