The bipartite analogue of a classical spanning tree enumeration formula, Boolean functions, and their applications to counting odd spanning trees

TL;DR

This paper introduces a new method for counting odd spanning trees in complete and bipartite graphs using classical enumeration formulas and Boolean functions, providing a simpler proof and generalization.

Contribution

It offers a novel, simplified proof for counting odd spanning trees and extends the enumeration to complete bipartite graphs.

Findings

Count of odd spanning trees in complete graphs derived

Extension of counting method to bipartite graphs

Simplified proof using classical formulas and Boolean functions

Abstract

Recently, Zheng and Wu defined the concept of odd spanning tree of a graph, meaning a spanning tree in which every vertex has odd degree. Similar to Cayley's formula, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Pr\"ufer code and the exponential generating function. In this note, we give a simple proof via a classical spanning tree enumeration formula and the Boolean function.We also generalize it to complete bipartite graphs.