arXiv subjects

Joseph E. Bonin

Publications and source records attributed to Joseph E. Bonin.

25 records · Page 2Linked to original sources

arXiv2009

A Note on the Sticky Matroid Conjecture

A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzik proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r at least 3, no rank-r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.

math.CO↗

arXiv2009

Splicing Matroids

We introduce and study a natural variant of matroid amalgams. For matroids M(A) and N(B) such that M/(A-B)=N(B-A), we define a splice of M and N to be a matroid L on the union of A and B with L(B-A)=M and L/(A-B)=N. We show that splices exist for each such pair of matroids M and N; furthermore, there is a freest splice of M and N, which we call the free splice. We characterize when a matroid L(E) is the free splice of L\U and L/V for subsets U and V of E. We study minors of free splices and the interaction between free splice and several other matroid operations. Although free splice is not an associative operation, we prove a weakened counterpart of associativity that holds in general and we characterize the triples for which associativity holds. We also study free splice as it relates to various classes of matroids.

math.CO↗

arXiv2007

The Lattice of Cyclic Flats of a Matroid

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice Z of sets and a function r on Z to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to Z. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids.

math.CO↗

arXiv2007

Transversal Lattices

A flat of a matroid is cyclic if it is a union of circuits; such flats form a lattice under inclusion and, up to isomorphism, all lattices can be obtained this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic flats are isomorphic to it are transversal. We investigate some sufficient conditions for a lattice to be a Tr-lattice; a corollary is that distributive lattices of dimension at most two are Tr-lattices. We give a necessary condition: each element in a Tr-lattice has at most two covers. We also give constructions that produce new Tr-lattices from known Tr-lattices.

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arXiv2004

Multi-Path Matroids

We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural properties. Key elements of this work are two complementary perspectives we develop for these matroids: on the one hand, multi-path matroids are transversal matroids that have special types of presentations; on the other hand, the bases of multi-path matroids can be viewed as sets of lattice paths in certain planar diagrams.

math.CO↗

arXiv2004

Lattice Path Matroids: Structural Properties

This paper studies structural aspects of lattice path matroids, a class of transversal matroids that is closed under taking minors and duals. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors.

math.CO↗

arXiv2002

Lattice path matroids: enumerative aspects and Tutte polynomials

Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m,r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the beta invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the beta invariant of certain lattice path matroids.

math.CO↗