Some inequalities for the Tutte polynomial

Abstract

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2011 ElsevierWe prove that the Tutte polynomial of a coloopless paving matroid is convex along the portion of the line x+y=p lying in the positive quadrant. Every coloopless paving matroid is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases. For this latter class we give a proof that TM(a,a)≤max{TM(2a,0),TM(0,2a)} for a≥2. We conjecture that TM(1,1)≤max{TM(2,0),TM(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.This study is partly supported by Conacyt of México Project 83977

Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials

“…Any sufficiently large number, like 8, will work. 2 To remind myself, with my memory failing in old age, let f (x) = (log x)(log log x) and g(x) = log x + log log x + log log log x + 1. Then f (17) ≥ g(17) > 0 and when we differentiate both functions, it is evident that log log…”

A Tutte polynomial inequality for lattice path matroids

“…An important result related to the multiplicative Merino-Welsh conjecture due to Jackson [13] is that T (M ; b, 0) · T (M ; 0, b) ≥ T (M ; a, a) 2 for any loopless, coloopless matroid M provided that b ≥ a(a + 2). Conjecture 1.2.1 for paving matroids, Catalan matroids, and whirls is proved in [7]. By combining the results from [7] and [13] it can be proved inductively that paving matroids even satisfy Conjecture 1.2.3 (the base cases need a detailed treatment).…”

“…Conjecture 1.2.1 for paving matroids, Catalan matroids, and whirls is proved in [7]. By combining the results from [7] and [13] it can be proved inductively that paving matroids even satisfy Conjecture 1.2.3 (the base cases need a detailed treatment).…”

“…Thus, arXiv:1510.00600v4 [math.CO] 27 Oct 2016 establishing Conjecture 1.1.1 in these cases. Chávez-Lomelí et al [7] proved Conjecture 1.1.1 for several families of graphs, including wheels and complete graphs. Noble and Royle [16] established Conjecture 1.1.3 for the class of series-parallel graphs.…”

The Merino–Welsh conjecture holds for series–parallel graphs

“…Chávez-Lomelí, Merino, Noble and Ramírez-Ibáñez [1] proved (among other results) that for a coloopless paving matroid M , the function T M (1 − x, 1 + x) is convex in the region −1 ≤ x ≤ 1, thus proving that paving matroids satisfy the Merino-Welsh conjecture.…”