Enumeration of Bipartite Graphs and Bipartite Blocks

Abstract

We use the theory of combinatorial species to count unlabelled bipartite graphs and bipartite blocks (nonseparable or 2-connected graphs). We start with bicolored graphs, which are bipartite graphs that are properly colored in two colors. The two-element group $\mathfrak{S}_2$ acts on these graphs by switching the colors, and connected bipartite graphs are orbits of connected bicolored graphs under this action. From first principles we compute the $\mathfrak{S}_2$-cycle index for bicolored graphs, an extension of the ordinary cycle index, introduced by Henderson, that incorporates the $\mathfrak{S}_2$-action. From this we can compute the $\mathfrak{S}_2$-cycle index for connected bicolored graphs, and then the ordinary cycle index for connected bipartite graphs. The cycle index for connected bipartite graphs allows us, by standard techniques, to count unlabeled bipartite graphs and bipartite blocks.