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Csikvári's poset and Tutte polynomial

Changxin Ding

TL;DR

The paper resolves a conjecture of Reiner and Smith by proving that among trees on $n$ vertices, the cone-Tutte polynomial $f(T)={\mathbf T}_{\text{Cone}(T)}(1,y)$ decreases coefficientwise along the Csikvári poset. The authors achieve this by leveraging Csikvári's General Lemma and introducing a compatible $g_v(G)$ function to factor the difference $f(T)-f(T')$ for covering relations into a product of nonnegative polynomials. This yields an affirmative strengthening of the extremal behavior initially observed for cones over trees and provides a new, equality-focused proof technique that reinforces the link between generalized tree shifts and Tutte polynomials. The results unify several prior extremal phenomena within a single poset framework and enhance understanding of how tree transformations influence cone-Tutte polynomials.

Abstract

Csikvári constructed a poset on trees to prove that several graph functions attain extreme values at the star and the path among the trees on a fixed number of vertices. Reiner and Smith proved that the Tutte polynomials $T(1,y)$ of cones over trees, which are the graphs obtained by attaching a cone vertex to a tree, have the described extreme behavior. They further conjectured that the result can be strengthened in terms of Csikvári's poset. We solve this conjecture affirmatively.

Csikvári's poset and Tutte polynomial

TL;DR

The paper resolves a conjecture of Reiner and Smith by proving that among trees on vertices, the cone-Tutte polynomial decreases coefficientwise along the Csikvári poset. The authors achieve this by leveraging Csikvári's General Lemma and introducing a compatible function to factor the difference for covering relations into a product of nonnegative polynomials. This yields an affirmative strengthening of the extremal behavior initially observed for cones over trees and provides a new, equality-focused proof technique that reinforces the link between generalized tree shifts and Tutte polynomials. The results unify several prior extremal phenomena within a single poset framework and enhance understanding of how tree transformations influence cone-Tutte polynomials.

Abstract

Csikvári constructed a poset on trees to prove that several graph functions attain extreme values at the star and the path among the trees on a fixed number of vertices. Reiner and Smith proved that the Tutte polynomials of cones over trees, which are the graphs obtained by attaching a cone vertex to a tree, have the described extreme behavior. They further conjectured that the result can be strengthened in terms of Csikvári's poset. We solve this conjecture affirmatively.
Paper Structure (5 sections, 6 theorems, 20 equations, 2 figures, 1 table)

This paper contains 5 sections, 6 theorems, 20 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tutte polynomial and Cone of Tree
  4. Csikvári poset
  5. Proof of Conjecture \ref{['conjecture']}

Key Result

Theorem 1.1

RS For a tree $T\in{\mathcal{T}}_n$, the inequalities hold coefficientwise.

Figures (2)

  • Figure 1: The Hasse diagram of the Csikvári poset on trees with $7$ vertices RS.
  • Figure 2: A generalized tree shift transforms a tree $T$ to anther tree $T'$ (Definition \ref{['treeshift']}).

Theorems & Definitions (17)

  • Theorem 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 3.1
  • Proposition 3.2
  • ...and 7 more