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

TL;DR

This paper generalizes Moon's formula to count spanning trees in complete bipartite graphs that contain a specific spanning forest, providing an explicit combinatorial count based on the forest's structure.

Contribution

The authors extend Moon's classic spanning tree counting formula from complete graphs to complete bipartite graphs, incorporating a fixed spanning forest.

Findings

Derived an explicit formula for counting spanning trees containing a given forest in $K_{m,n}$

Generalized Moon's formula to bipartite graphs

Provided a combinatorial expression involving the forest's component sizes

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 $K_{m,n}$ with $|X|=m$ and $|Y|=n$. We prove that for any given spanning forest $F$ of $K_{m,n}$ with components $T_1,T_2,\ldots,T_k$, the number of spanning trees in $K_{m,n}$ which contain all edges in $F$ is equal to $$ \frac 1{mn}\left(\prod_{i=1}^k (m_in+n_im)\right) \left (1-\sum_{i=1}^{k}\frac{m_in_i}{m_in+n_im}\right ), $$ where $m_i=|V(T_i)\cap X|$ and $n_i=|V(T_i)\cap Y|$ for $i=1,2,\ldots,k$.