Enumeration of spanning trees of complete multipartite graphs containing a fixed spanning forest

Abstract

AbstractMoon's classical result implies that the number of spanning trees of a complete graph with vertices containing a given spanning forest equals , where is the number of components of , and are the numbers of vertices of component of . Dong and Ge extended this result to the complete bipartite graph, and obtain an interesting formula to count spanning trees of a complete bipartite graph containing a given spanning forest . They also posed the problem to count spanning trees of a complete ‐partite graph containing a given spanning forest for . In this paper, we propose a technique to solve this problem. Using this technique, we obtain closed formulae to count spanning trees of complete ‐partite graphs containing a given spanning forest for and 4. Our technique also results in a new and simple proof of Dong and Ge's formula.