Lagrangian Matroids: Representations of Type BnB_n

We introduce the concept of orientation for Lagrangian matroids represented in the flag variety of maximal isotropic subspaces of dimension N in the real vector space of dimension 2N+1. The paper continues the study started in math.CO/0209100.Comment: Requires amssymb.sty; 17 page

The Interlace Polynomial

In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic systems and delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials, edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL

Note on Hamiltonicity of basis graphs of even delta-matroids

We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges $e$ and $f$ sharing a common end, it has a Hamiltonian cycle using $e$ and avoiding $f$ unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length $\ell\ge 3$, and each edge belongs to cycles of every length $\ell \ge 4$. For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).Comment: 10 pages, 2 figures. Corrected a typ

A Min-Max . . . Functions and Its Implications

A. Huber and V. Kolmogorov (ISCO 2012) introduced a concept of k-submodular function as a generalization of ordinary submodular (set) functions and bisubmodular functions and obtained a min-max theorem for minimization of k-submodular functions. Also F. Kuivinen (2011) considered submodular functions on (product lattices of) diamonds and showed a min-max theorem for minimization of submodular functions on diamonds. In the present paper we consider a common generalization of k-submodular functions and submodular functions on diamonds, which we call a transversal submodular function (or a t-submodular function, for short). We show a min-max theorem for minimization of t-submodular functions in terms of a new norm composed of ℓ1 and ℓ ∞ norms. This reveals a relationship between the obtained min-max theorem and that for minimization of ordinary submodular set functions due to J. Edmonds (1970). We also show how our min-max theorem for t-submodular functions can be used to prove the min-max theorem for k-submodular functions by Huber and Kolmogorov and that for submodular functions on diamonds by Kuivinen. Moreover, we show a counterexample to a characterization, given by Huber and Kolmogorov (ISCO 2012), of extreme points of the k-submodular polyhedron and make it a correct one by fixing a flaw therein

Rank functions and invariants of delta-matroids

In this note, we give a rank function axiomatization for delta-matroids and study the corresponding rank generating function. We relate an evaluation of the rank generating function to the number of independent sets of the delta-matroid, and we prove a log-concavity result for that evaluation using the theory of Lorentzian polynomials

Isotropical Linear Spaces and Valuated Delta-Matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D