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Degree Sequences vs. Forests in Bipartite Graphs

Darij Grinberg, Benjamin Liber

Abstract

We prove a conjecture of Shteiner and Shteyner stating that for a bipartite graph $G=(V,E)$, the number of forests in $G$ equals the number of degree sequences arising from its spanning subgraphs. In the process, we provide several equivalent evaluations of the Tutte polynomial $T_G(x,y)$ at $(2,1)$, including interpretations in terms of degree vectors obtained from orientations of $G$.

Degree Sequences vs. Forests in Bipartite Graphs

Abstract

We prove a conjecture of Shteiner and Shteyner stating that for a bipartite graph , the number of forests in equals the number of degree sequences arising from its spanning subgraphs. In the process, we provide several equivalent evaluations of the Tutte polynomial at , including interpretations in terms of degree vectors obtained from orientations of .
Paper Structure (6 sections, 12 theorems, 62 equations)

This paper contains 6 sections, 12 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Tools and Lemmas
  3. The Tutte Polynomial
  4. Orientations of Graphs
  5. Equivalent Descriptions of $T_G\left( 2,1 \right)$
  6. Proof of Theorem \ref{['maintheorem']} and Further Remarks

Key Result

Theorem 1.2

If $G=(V,E)$ is bipartite, then

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 18 more