Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials

TL;DR

The paper investigates the Merino-Welsh conjecture relating Tutte polynomial evaluations $T(M;2,0)$, $T(M;0,2)$, and $T(M;1,1)$ for matroids, focusing on identifying sufficient conditions under which $T(M;2,0)+T(M;0,2) \ge 2\,T(M;1,1)$ holds for matroids with no loops or isthmuses. It provides two broad sufficient conditions: a density-based bound showing the inequality for isthmus-free matroids with rank $r$ and size $n$ satisfying $n \ge \lceil r(\log_2 r + \log_2\log_2 r + \log_2\log_2\log_2 r) \rceil$, and a cocircuit-size bound where every cocircuit has size at least $r+1$, with notes on refinements and related loop-containing cases. The methods hinge on interpreting $T(M;2,0)$ and $T(M;0,2)$ as counts of combinatorial objects, using binomial estimates $T(M;1,1) \le \binom{n}{r}$, and applying a forward-difference/monotonicity argument to compare growth of $2^{n-r}$ and $\binom{n}{r}$. These results are framed as insubstantial in resolving the full conjecture but illuminate tractable regimes and suggest directions where sharper bounds or asymptotics might extend the reach of Merino-Welsh.

Abstract

The Merino-Welsh conjectures say that subject to conditions, there is an inequality among the Tutte-polynomial evaluations $T(M;2,0)$, $T(M;0,2)$, and $T(M;1,1)$. We present three results on a Merino-Welsh conjecture. These results are "inconsequential" in the sense that although they imply a version of the conjecture for many matroids, they seem to be dead ends.

Theorems & Definitions (8)

• Conjecture 1.1
• Theorem 2.1
• Proposition 2.2
• Lemma 2.3
• Theorem 3.1
• Lemma 3.2
• Proposition 3.3
• proof