Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Around the Merino--Welsh conjecture: improving Jackson's inequality

Péter Csikvári

TL;DR

The paper addresses the Merino–Welsh conjecture and its matroid generalization by establishing sharper multiplicative inequalities for the Tutte polynomial via the permutation Tutte polynomial. It develops a framework around the local basis exchange graph and Harris’ inequality to convert local bounds into global ones, yielding an improved constant x=2.355 for which T_M(x,0)T_M(0,x) ≥ T_M(1,1)^2 holds for all loopless and coloopless matroids, and further shows that matroids with prescribed circuit-length ranges (in M and M^*) satisfy the conjecture. The results substantially broaden the known cases where Merino–Welsh holds and provide a systematic, computation-backed approach that could extend to other polynomial inequalities. This approach bridges combinatorial polynomial inequalities and probabilistic inequalities on bipartite graphs, with potential implications for related conjectures and matroid classes.

Abstract

The Merino-Welsh conjecture states that for a graph $G$ without loops and bridges we have $$\max(T_G(2,0),T_G(0,2))\geq T_G(1,1).$$ Later Jackson proved that for any matroid $M$ without loop and coloop we have $$T_M(3,0)T_M(0,3)\geq T_M(1,1)^2.$$ The value $3$ in this statement was improved to $2.9242$ by Beke, Csáji, Csikvári and Pituk. In this paper, we further improve on this result by showing that $$T_M(2.355,0)T_M(0,2.355)\geq T_M(1,1)^2.$$ We also prove that the Merino--Welsh conjecture is true for matroids $M$, where all circuits of $M$ and its dual $M^*$ have length between $\ell$ and $(\ell-2)^4$ for some $\ell\geq 6$.

Around the Merino--Welsh conjecture: improving Jackson's inequality

TL;DR

The paper addresses the Merino–Welsh conjecture and its matroid generalization by establishing sharper multiplicative inequalities for the Tutte polynomial via the permutation Tutte polynomial. It develops a framework around the local basis exchange graph and Harris’ inequality to convert local bounds into global ones, yielding an improved constant x=2.355 for which T_M(x,0)T_M(0,x) ≥ T_M(1,1)^2 holds for all loopless and coloopless matroids, and further shows that matroids with prescribed circuit-length ranges (in M and M^*) satisfy the conjecture. The results substantially broaden the known cases where Merino–Welsh holds and provide a systematic, computation-backed approach that could extend to other polynomial inequalities. This approach bridges combinatorial polynomial inequalities and probabilistic inequalities on bipartite graphs, with potential implications for related conjectures and matroid classes.

Abstract

The Merino-Welsh conjecture states that for a graph without loops and bridges we have Later Jackson proved that for any matroid without loop and coloop we have The value in this statement was improved to by Beke, Csáji, Csikvári and Pituk. In this paper, we further improve on this result by showing that We also prove that the Merino--Welsh conjecture is true for matroids , where all circuits of and its dual have length between and for some .

Paper Structure

This paper contains 14 sections, 21 theorems, 122 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tutte polynomial and matroids
  4. Permutation Tutte polynomial
  5. Warm-up: asymptotic computation of some permutation Tutte polynomial
  6. The main lemma
  7. Proof of Theorem \ref{['main theorem']}.
  8. Matroids with prescribed circuit lengths
  9. Concluding remarks
  10. Appendix
  11. Table for Idea 1.
  12. Table for Idea 2.
  13. Table for Idea 3.
  14. Table for Idea 4.

Key Result

Theorem 1.1

There are infinitely many matroids $M$ without loops and coloops for which Furthermore, let $x_0$ be the largest root of the polynomial $x^3-9(x-1)$. ($x_0\approx 2.22668...$) Then for $0\leqslant a<x_0$ there are infinitely many matroids $M$ without loops and coloops for which

Figures (2)

  • Figure 1: Example for a graph $G$ and the local basis exchange graph $H[T]$ obtained from a spanning tree $T$.
  • Figure 2: The graph $H_{6,6,6}$.

Theorems & Definitions (41)

  • Theorem 1.1: Beke, Csáji, Csikvári and Pituk beke2024merino
  • Theorem 1.2: Jackson jackson2010inequality
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: beke2024permutation
  • Theorem 2.2: Tutte tutte1954contribution
  • Definition 2.3
  • Lemma 2.4: Beke, Csáji, Csikvári, Pituk beke2024permutation
  • proof
  • Remark 2.5
  • ...and 31 more