36 research outputs found
Some inequalities for the Tutte polynomial
We prove that the Tutte polynomial of a coloopless paving matroid is convex
along the portions of the line segments x+y=p lying in the positive quadrant.
Every coloopless paving matroids is in the class of matroids which contain two
disjoint bases or whose ground set is the union of two bases of M*. For this
latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a
>= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same
class of matroids. We also prove this conjecture for some families of graphs
and matroids.Comment: 17 page
Hopf algebras and Tutte polynomials
By considering Tutte polynomials of Hopf algebras, we show how a Tutte
polynomial can be canonically associated with combinatorial objects that have
some notions of deletion and contraction. We show that several graph
polynomials from the literature arise from this framework. These polynomials
include the classical Tutte polynomial of graphs and matroids, Las Vergnas'
Tutte polynomial of the morphism of matroids and his Tutte polynomial for
embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal
polynomial, and the Penrose polynomial.
We show that our Tutte polynomials of Hopf algebras share common properties
with the classical Tutte polynomial, including deletion-contraction
definitions, universality properties, convolution formulas, and duality
relations. New results for graph polynomials from the literature are then
obtained as examples of the general results.
Our results offer a framework for the study of the Tutte polynomial and its
analogues in other settings, offering the means to determine the properties and
connections between a wide class of polynomial invariants.Comment: v2: change of title and some reorderin
Universal Tutte characters via combinatorial coalgebras
The Tutte polynomial is the most general invariant of matroids and graphs
that can be computed recursively by deleting and contracting edges. We
generalize this invariant to any class of combinatorial objects with deletion
and contraction operations, associating to each such class a universal Tutte
character by a functorial procedure. We show that these invariants satisfy a
universal property and convolution formulae similar to the Tutte polynomial.
With this machinery we recover classical invariants for delta-matroids, matroid
perspectives, relative and colored matroids, generalized permutohedra, and
arithmetic matroids, and produce some new convolution formulae. Our principal
tools are combinatorial coalgebras and their convolution algebras. Our results
generalize in an intrinsic way the recent results of
Krajewski--Moffatt--Tanasa.Comment: Accepted version, 51p
A free subalgebra of the algebra of matroids
This paper is an initial inquiry into the structure of the Hopf algebra of
matroids with restriction-contraction coproduct. Using a family of matroids
introduced by Crapo in 1965, we show that the subalgebra generated by a single
point and a single loop in the dual of this Hopf algebra is free.Comment: 19 pages, 3 figures. Accepted for publication in the European Journal
of Combinatorics. This version incorporates a few minor corrections suggested
by the publisher