Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.05722

A sharp upper bound on the number of spanning forests of regular graphs

T. Wu·S. Lu·X. Dong
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $G$ be a simple graph on $n$ vertices, and let $F(G)$ denote the number of its spanning forests. Bencs and Csikvári [Upper bound for the number of spanning forests of regular graphs, European J. Combin. 110 (2023) 103677] proved that every $r$-regular graph $G$ with $r\geq 2$ satisfies $F(G) \leq r^{n}$. They further conjectured that for $r \geq 3$, \[ F(G)^{1/n} \leq \frac{(r - 1)^{r-1}}{(r^2 - 2r - 1)^{r/2-1}}. \] In this paper, we resolve this conjecture in the affirmative.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

T. Wu, S. Lu, X. Dong. 2026-10-05. A sharp upper bound on the number of spanning forests of regular graphs. https://arxiv.org/abs/2610.05722

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Twin-star hypothesis and cycle-free $d$-partitions of $K_{2d}$

In this paper we study an equivalence relation defined on the set of cycle-free $d$-partitions of the complete graph $K_{2d}$. We discuss a conjecture which states that this equivalence relation has only one equivalence class, and show that the conjecture is equivalent with the so called twin-star hypothesis. We check the conjecture in the case $d=4$ and disuses how this relates to the determinant-like map $det^{S^2}$.

math.CO↗
arXiv2026

Oriented trees in digraphs without short non-directed cycles

The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell+1$ and minimum degree at least $k/\ell$ contains every tree with $k$ edges whose maximum degree does not exceed the minimum degree of $G$. In this paper, we extend Jiang's result to digraphs by proving that every digraph $D$ with no non-directed cycle of length between $3$ and $2\ell$ and minimum semidegree at least $k/\ell$ contains every oriented tree with $k$ edges whose maximum degree does not exceed the minimum semidegree of $D$. This answers a question raised by Stein and Trujillo-Negrete in the affirmative.

math.CO↗
arXiv2026

The 3-state Potts model on planar triangulations: explicit algebraic solution

We consider the $3$-state Potts generating function $T(ν,w)$ of planar triangulations; that is, the bivariate series that counts planar triangulations with vertices coloured in $3$ colours, weighted by their size (number of vertices, recorded by the variable $w$) and by the number of monochromatic edges (variable $ν$). This series was proved to be algebraic 15 years ago by Bernardi and the first author: this follows from its link with the solution of a discrete differential equation (DDE), and from general algebraicity results on such equations. However, despite recent progresses on the effective solution of DDEs, the exact value of $T(ν,w)$ has remained unknown so far -- except in the case $ν=0$, corresponding to proper colourings and solved by Tutte in the sixties. We determine here this exact value, proving that $T(ν,w)$ satisfies a polynomial equation of degree $11$ in $T$ and genus $1$ in $w$ and $T$. We prove that the critical value of $ν$ is $ν_c=1+3/\sqrt{47}$, with a critical exponent $6/5$ in the series $T(ν_c, \cdot)$, while the other values of $ν$ yield the usual map exponent $3/2$. By duality of the planar Potts model, our results also characterize the 3-state Potts generating function of planar cubic maps, in which all vertices have degree $3$. In particular, the annihilating polynomial, still of degree $11$, that we obtain for properly 3-coloured cubic maps proves a conjecture by Bruno Salvy from 2009.

math.CO↗