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

Jun Ge, Yamin Yu

TL;DR

The paper addresses counting odd spanning trees in graphs by deriving a bipartite analogue of a classical degree-sequence spanning-tree formula and coupling it with a Boolean-function moment argument to obtain closed forms. It provides explicit formulas for tau_o(K_n) and tau_o(K_{m,n}) and shows parity-based zeros, illustrating a simple, unified approach that connects Boolean-methodology with traditional graph enumeration. The results extend the known complete-graph case and generalize to complete bipartite graphs, offering a compact framework for counting odd spanning trees. This work highlights a direct link between Boolean function techniques and classical spanning-tree enumeration.

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 Prüfer 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.

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

TL;DR

The paper addresses counting odd spanning trees in graphs by deriving a bipartite analogue of a classical degree-sequence spanning-tree formula and coupling it with a Boolean-function moment argument to obtain closed forms. It provides explicit formulas for tau_o(K_n) and tau_o(K_{m,n}) and shows parity-based zeros, illustrating a simple, unified approach that connects Boolean-methodology with traditional graph enumeration. The results extend the known complete-graph case and generalize to complete bipartite graphs, offering a compact framework for counting odd spanning trees. This work highlights a direct link between Boolean function techniques and classical spanning-tree enumeration.

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 Prüfer 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.
Paper Structure (4 sections, 6 theorems, 22 equations)

This paper contains 4 sections, 6 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bipartite analogue of a classical spanning tree enumeration formula
  4. Applications to counting odd spanning trees

Key Result

Theorem 1.1

Let $n$ be a positive number. Then

Theorems & Definitions (10)

  • Theorem 1.1: FCW
  • Lemma 1
  • proof
  • Theorem 3.1: BergeLovasz
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof