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Counting Isomorphism Classes of Spanning Trees of Complete Bipartite Graphs

Peter Johnson, Shayne Nochumson

TL;DR

The paper addresses counting the number $I_{a,b}$ of isomorphism classes of spanning trees in the complete bipartite graph $K_{a,b}$ for $2 \le a \le b$. It introduces a partition-based degree-sequence encoding via pairs $(s_i),(t_j)$ of partitions of $a+b-1$ and constructs connected bipartite trees $Q((s_i),(t_j))$ that realize these degrees, proving every spanning tree of $K_{a,b}$ arises this way. This yields a universal lower bound $I_{a,b} \ge P_a(a+b-1) \cdot P_b(a+b-1)$, with stronger refinement in the square case $a=b$ giving $I_{a,a} \ge \frac{r(r+1)}{2}$ where $r = P_a(2a-1)$. The results situate the isomorphism-class count between partition-theoretic quantities and classical labeled-span­ning-tree counts (via Cayley/Scoins), highlighting a combinatorial bridge between graph structure and partition theory. Overall, the work provides explicit, unlabeled lower bounds on spanning-tree isomorphism classes and clarifies their relationship to known enumeration bounds in bipartite graphs.

Abstract

Spanning trees of complete bipartite graphs exhibit a rich interaction between degree sequences and graph structure. In this paper, we obtain lower bounds on the number of isomorphism classes of spanning trees in $K_{a,b}, 2 \leq a \leq b$ in terms of $P_a(a+b-1)$ and $P_b(a+b-1)$ where $P_k(m)$ is the number of integer partitions of $m$ of length $k$.

Counting Isomorphism Classes of Spanning Trees of Complete Bipartite Graphs

TL;DR

The paper addresses counting the number of isomorphism classes of spanning trees in the complete bipartite graph for . It introduces a partition-based degree-sequence encoding via pairs of partitions of and constructs connected bipartite trees that realize these degrees, proving every spanning tree of arises this way. This yields a universal lower bound , with stronger refinement in the square case giving where . The results situate the isomorphism-class count between partition-theoretic quantities and classical labeled-span­ning-tree counts (via Cayley/Scoins), highlighting a combinatorial bridge between graph structure and partition theory. Overall, the work provides explicit, unlabeled lower bounds on spanning-tree isomorphism classes and clarifies their relationship to known enumeration bounds in bipartite graphs.

Abstract

Spanning trees of complete bipartite graphs exhibit a rich interaction between degree sequences and graph structure. In this paper, we obtain lower bounds on the number of isomorphism classes of spanning trees in in terms of and where is the number of integer partitions of of length .
Paper Structure (2 sections, 6 theorems, 1 figure)

This paper contains 2 sections, 6 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Lemma 2.1

Let $2 \leq a \leq b$ and $a+b=n$. For any pair of integer partitions $s_1 \geq ... \geq s_a >0$ and $t_1 \geq ... \geq t_b >0$ of $a+b-1$, there is a connected bipartite graph $Q = Q((s_i),(t_j))$ with bipartitions $A,B$ where $|A|=a, |B|=b$ and where the $s_i$ are the degrees in $Q$ of vertices in

Figures (1)

  • Figure 1: $Q((2,1),(2,1))$

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.3.1
  • Theorem 2.4
  • proof
  • Corollary 2.4.1