Degree Sequences vs. Forests in Bipartite Graphs

TL;DR

This paper proves a conjecture linking the number of forests in bipartite graphs to degree sequences of their spanning subgraphs, and provides new evaluations of the Tutte polynomial at specific points.

Contribution

It establishes the equality between forests count and degree sequences in bipartite graphs and offers novel interpretations of the Tutte polynomial at (2,1).

Findings

Number of forests equals the count of degree sequences from spanning subgraphs.

Provides equivalent evaluations of the Tutte polynomial at (2,1).

Interprets Tutte polynomial evaluations via orientations and degree vectors.

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 $G$.