Counting spanning trees in a complete bipartite graph which contain a given spanning forest

Abstract

In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let (X, Y ) be the bipartition of the complete bipartite graph Km,n with |X| = m and |Y | = n. We prove that for any given spanning forest F of Km,n with components T1, T2, • • • , T k , the number of spanning trees in Km,n which contain all edges in F is equal to 1 mn k i=1 (min + nim) 1 − k i=1 mini min + nim , where mi = |V (Ti) ∩ X| and ni = |V (Ti) ∩ Y | for i = 1, 2, • • • , k.