On minimum-size matroid counterexamples to the multiplicative Merino--Welsh inequality
Abstract
Let M be a matroid and let TutteM(x,y) be its Tutte polynomial. The multiplicative Merino--Welsh inequality TutteM(2,0)TutteM(0,2)≥ TutteM(1,1)2 was proved to be false for matroids by Beke, Csáji, Csikvári, and Pituk (\emph{Adv. Math.} \textbf{446} (2024), 109674). Let Un,r denote the rank-r uniform matroid on n elements, where 0<r<n, and let Un,r(q) denote the matroid obtained by replacing each element of Un,r with q parallel copies. They constructed an infinite family of counterexamples of the form Un,r(2), including U33,22(2). In this paper, we prove that every Un,r(q) with q≥3 satisfies the inequality. For q=2, we prove that the minimum number of elements in such a counterexample is $62$, attained by U31,21(2). For loopless and bridgeless graphs, Noble and Royle (\emph{Eur. J. Comb.} \textbf{38} (2014), 24--35) showed that every edge of a minimum-size counterexample to the multiplicative Merino--Welsh conjecture lies in a series or parallel class of size two or three. For matroids, we further prove that every element of a minimum-size loopless and coloopless counterexample lies in exactly one series or parallel class of size two.
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BibTeX
@misc{chen2026minimumsize,
title = {{On minimum-size matroid counterexamples to the multiplicative Merino--Welsh inequality}},
author = {Jungang Chen and Xian'an Jin and Tianlong Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27706},
eprint = {2610.27706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27706},
}
RIS
TY - UNPB
TI - On minimum-size matroid counterexamples to the multiplicative Merino--Welsh inequality
AU - Jungang Chen
AU - Xian'an Jin
AU - Tianlong Ma
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27706
UR - https://arxiv.org/abs/2610.27706
AB - Let M be a matroid and let Tutte_M(x,y) be its Tutte polynomial. The multiplicative Merino--Welsh inequality Tutte_M(2,0)Tutte_M(0,2)≥ Tutte_M(1,1)^2 was proved to be false for matroids by Beke, Csáji, Csikvári, and Pituk (\emph{Adv. Math.} \textbf{446} (2024), 109674). Let U_n,r denote the rank-r uniform matroid on n elements, where 0<r<n, and let U_n,r^(q) denote the matroid obtained by replacing each element of U_n,r with q parallel copies. They constructed an infinite family of counterexamples of the form U_n,r^(2), including U_33,22^(2). In this paper, we prove that every U_n,r^(q) with q≥3 satisfies the inequality. For q=2, we prove that the minimum number of elements in such a counterexample is $62$, attained by U_31,21^(2). For loopless and bridgeless graphs, Noble and Royle (\emph{Eur. J. Comb.} \textbf{38} (2014), 24--35) showed that every edge of a minimum-size counterexample to the multiplicative Merino--Welsh conjecture lies in a series or parallel class of size two or three. For matroids, we further prove that every element of a minimum-size loopless and coloopless counterexample lies in exactly one series or parallel class of size two.
ER -