Criel Merino

Graph Polynomials and Their Applications I: The Tutte Polynomial

In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomials many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized, and how they can encode information relevant to physical applications. We conclude with a brief discussion of computational complexity considerations.

Graph Polynomials and Their Applications II: Interrelations and Interpretations

This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial information it contains. The polynomials we discuss here are not generally specializations of the Tutte polynomial, but they are each in some way related to the Tutte polynomial, and often to one another. We emphasize these interrelations and explore how an understanding of one polynomial can guide research into others. We also discuss multivariable generalizations of some of these polynomials and the theory facilitated by this. We conclude with two examples, one from biology and one from physics, that illustrate the applicability of graph polynomials in other fields. This is the second chapter of a two chapter series, and concludes Graph Polynomials and Their Applications I: The Tutte Polynomial, arXiv:0803.3079

The chip-firing game

The process called the chip-firing game has been around for no more than 20 years, but it has rapidly become an important and interesting object of study in structural combinatorics. The reason for this is partly due to its relation with the Tutte polynomial and group theory, but also because of the contribution of people in theoretical physics who know it as the (Abelian) sandpile model. Here, we survey some of the numerous connections that the chip-firing game has with some other parts of combinatorics and with theoretical physics. Among these we present its relation with the Tutte polynomial, group theory, greedoids with repetition and matroids. We also reintroduce it as the Abelian sandpile model of statistical mechanics and give a relation with the Potts model.

On the structure of the h-vector of a paving matroid

We give two proofs that the h-vector of any paving matroid is a pure O-sequence, thus answering in the affirmative a conjecture made by Stanley, for this particular class of matroids. We also investigate the problem of obtaining good lower bounds for the number of bases of a paving matroid given its rank and number of elements.

The Equivalence of Two Graph Polynomials And a Symmetric Function

The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial, due to Stanley, are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to strengthen them, which also captures Tuttes universal V-function as a specialization. We show that the equivalence remains true for the strong functions, thus answering a question raised by Dominic Welsh.

Note: On the length of longest alternating paths for multicoloured point sets in convex position

Let P be a set of points in R^2 in general position such that each point is coloured with one of k colours. An alternating path of P is a simple polygonal whose edges are straight line segments joining pairs of elements of P with different colours. In this paper we prove the following: suppose that each colour class has cardinality s and P is the set of vertices of a convex polygon. Then P always has an alternating path with at least (k-1)s elements. Our bound is asymptotically sharp for odd values of k.

On plane spanning trees and cycles of multicolored point sets with few intersections

Let P1, ..., Pk be a collection of disjoint point sets in R2 in general position. We prove that for each 1 ≤ i ≤ k we can find a plane spanning tree Ti of Pi such that the edges of T1 ,..., Tk intersect at most (k - 1)(n - k) + ½k(k - 1), where n is the number of points in P1 ∪ ... ∪ Pk. If the intersection of the convex hulls of P1,...., Pk is nonempty, we can find k spanning cycles such that their edges intersect at most (k - 1)n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 8n times, where |P ∪ Q| = n (the weight of an edge is its length).

Simple Euclidean Arrangements with No (≥ 5)-Gons

It is shown that if a simple Euclidean arrangement of n pseudolines has no (≥ 5)-gons, then it has exactly n - 2 triangles and (n - 2)(n - 3)/2 quadrilaterals. We also describe how to construct all such arrangements, and as a consequence we show that they are all stretchable.

A Problem on Hinged Dissections with Colours

Abstract.We examine the following problem. Given a square C we want a hinged dissection of C into congruent squares and a colouring of the edges of these smaller squares with k colours such that we can transform the original square into another with its perimeter coloured with colour i, for all i in We have the restriction that the moves have to be realizable in the plane, so when swinging the pieces no overlappings are allowed. We show a solution for k colours that uses p2 pieces, with p an even number and at least this by using a necklace made of the p2 pieces and an ingenious way to wrap it into a square.

On the Intersection Number of Matchings and Minimum Weight Perfect Matchings of Multicolored Point Sets

Let P and Q be disjoint point sets with 2r and 2s elements respectively, and M1 and M2 be their minimum weight perfect matchings (with respect to edge lengths). We prove that the edges of M1 and M2 intersect at most |M1|+|M2|−1 times. This bound is tight. We also prove that P and Q have perfect matchings (not necessarily of minimum weight) such that their edges intersect at most min{r,s} times. This bound is also sharp.