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<p>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 prove the bipartite analogue of a classical spanning tree enumeration formula. By using them and the Boolean function, we give simple proofs for the number of odd spanning trees of complete graphs and complete bipartite graphs.</p>

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

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Jun Ge,
  • Yamin Yu

摘要

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 prove the bipartite analogue of a classical spanning tree enumeration formula. By using them and the Boolean function, we give simple proofs for the number of odd spanning trees of complete graphs and complete bipartite graphs.