Products of Foldable Triangulations

Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special attention is paid to the cube case.Comment: new title; several paragraphs reformulated; 23 page

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction

Colorful Associahedra and Cyclohedra

Every n-edge colored n-regular graph G naturally gives rise to a simple abstract n-polytope, the colorful polytope of G, whose 1-skeleton is isomorphic to G. The paper describes colorful polytope versions of the associahedron and cyclohedron. Like their classical counterparts, the colorful associahedron and cyclohedron encode triangulations and flips, but now with the added feature that the diagonals of the triangulations are colored and adjacency of triangulations requires color preserving flips. The colorful associahedron and cyclohedron are derived as colorful polytopes from the edge colored graph whose vertices represent these triangulations and whose colors on edges represent the colors of flipped diagonals.Comment: 21 pp, to appear in Journal Combinatorial Theory

On multidimensional consistent systems of asymmetric quad-equations

Multidimensional Consistency becomes more and more important in the theory of discrete integrable systems. Recently, we gave a classification of all 3D consistent 6-tuples of equations with the tetrahedron property, where several novel asymmetric systems have been found. In the present paper we discuss higher-dimensional consistency for 3D consistent systems coming up with this classification. In addition, we will give a classification of certain 4D consistent systems of quad-equations. The results of this paper allow for a proof of the Bianchi permutability among other applications.Comment: 16 pages, 17 figure

The Order Dimension of the Poset of Regions in a Hyperplane Arrangement

We show that the order dimension of the weak order on a Coxeter group of type A, B or D is equal to the rank of the Coxeter group, and give bounds on the order dimensions for the other finite types. This result arises from a unified approach which, in particular, leads to a simpler treatment of the previously known cases, types A and B. The result for weak orders follows from an upper bound on the dimension of the poset of regions of an arbitrary hyperplane arrangement. In some cases, including the weak orders, the upper bound is the chromatic number of a certain graph. For the weak orders, this graph has the positive roots as its vertex set, and the edges are related to the pairwise inner products of the roots.Comment: Minor changes, including a correction and an added figure in the proof of Proposition 2.2. 19 pages, 6 figure

Acyclic orientations with path constraints

Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-defining and the introduction of new classes of valid inequalities