Upper bound for the number of spanning forests of regular graphs

TL;DR

This paper establishes an upper bound on the number of spanning forests in regular graphs, improving previous bounds and proposing a conjecture for the optimal bound based on the degree of the graph.

Contribution

The paper introduces a new upper bound for spanning forests in regular graphs and conjectures the exact asymptotic maximum, refining prior estimates.

Findings

New upper bound: $F(G) \,\leq\, d^n$

Improved bound over previous work by Kahale and Schulman

Conjecture for the optimal bound based on degree $d$

Abstract

We show that if $G$ is a $d$--regular graph on $n$ vertices, then the number of spanning forests $F(G)$ satisfies $F(G)\leq d^n$. The previous best bound due to Kahale and Schulman gave $(d+1/2+O(1/d))^n$. We also have the more precise conjecture that $$F(G)^{1/n}\leq \frac{(d-1)^{d-1}}{(d^2-2d-1)^{d/2-1}}.$$ If this conjecture is true, then the expression on the right hand side is the best possible.