A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results.Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra.     

Visibility Graphs and Oriented Matroids

   Abstract. We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.     

Visibility Graphs and Oriented Matroids

Abstract. We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.     

Spatial Graphs and Oriented Matroids: the Trefoil

Let m=m(K) be the smallest positive integer such that every linear spatial representation of the complete graph with n vertices, n ≥ m , contains either the knot K or its mirror . In this paper we show that m(Trefoil)=7 . The proof uses the theory of oriented matroids. Received July 8, 1997, and in revised form January 16, 1998.     

Desargues Theorem, Dynamics, and Hyperplane Arrangements

The Desargues theorem is a basic theorem in classical projective geometry. In this paper we generalize Desargues theorem in the direction of dynamical systems. Our result comprises an infinite family of configurations, having unbounded complexity. The proof of the result involves constructing special kinds of hyperplane arrangements and then projecting subsets of them into the plane.     

Piles of Cubes, Monotone Path Polytopes, and Hyperplane Arrangements

Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements and geometry of monotone path polytopes. Received January 24, 1997, and in revised form April 8, 1997.     

Intersection and Linking Numbers in Oriented Matroids

We introduce for oriented matroids a generalization of the concepts of intersection and linking numbers in Euclidean space, with most of their main properties (see Wu). As an application, we reprove a result of Brehm in a slightly extended form.     

On the Heterochromatic Number of Hypergraphs Associated to Geometric Graphs and to Matroids

The heterochromatic number h _c (H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We denote by ν(H) the number of vertices of H and by τ(H) the size of the smallest set containing at least two vertices of each hyperedge of H. For a complete geometric graph G with n ≥ 3 vertices let H = H(G) be the hypergraph whose vertices are the edges of G and whose hyperedges are the edge sets of plane spanning trees of G. We prove that if G has at most one interior vertex, then h _c (H) = ν(H) ? τ(H) + 2. We also show that h _c (H) = ν(H) ? τ(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given by the ground set and the bases of a matroid, respectively.     

Bernoulli Trials of Fixed Parity,Random and Randomly Oriented Graphs

Suppose you can colour n biased coins with n colours, all coins having the same bias. It is forbidden to colour both sides of a coin with the same colour, but all other colourings are allowed. Let X be the number of different colours after a toss of the coins. We present a method to obtain an upper bound on a median of X. Our method is based on the analysis of the probability distribution of the number of vertices with even in-degree in graphs whose edges are given random orientations. Our analysis applies to the distribution of the number of vertices with odd degree in random sub-graphs of fixed graphs. It turns out that there are parity restrictions on the random variables that are under consideration. Hence, in order to present our result, we introduce a class of Bernoulli random variables whose total number of successes is of fixed parity and are closely related to Poisson trials conditional on the event that their outcomes have fixed parity.     

二部定向图的最长通路和圈

Let D be a bipartite oriented graph in which the indegree and outdegree of each vertex are at least k. The result given in this paper is that D contains either a cycle of length at least 4k or a path of length at least 4k + 1. Jackson [1] declared that: if |V(D) |≤4k,D contains a Hamiltonian cycle. Evidently, the result Of this paper implies the result given by Jackson.     

Orientations, Lattice Polytopes, and Group Arrangements I: Chromatic and Tension Polynomials of Graphs

This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such a viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial T(G; x, y) of a graph G at (1, 0) as the number of cut-equivalence classes of acyclic orientations on G.     

Self-Converse and Oriented Graphs among the Third Parts of Nearly Complete Digraphs

H into t isomorphic parts is generalized so that either a remainder R or a surplus S, both of the numerically smallest possible size, are allowed. The sets of such nearly parts are defined to be the floor class and the ceiling class , respectively. We restrict ourselves to the case of nearly third parts of , the complete digraph, with . Then if , else and . The existence of nearly third parts which are oriented graphs and/or self-converse digraphs is settled in the affirmative for all or most n's. Moreover, it is proved that floor classes with distinct R's can have a common member. The corresponding result on the nearly third parts of the complete 2-fold graph is deduced. Furthermore, also if . Received: September 12, 1994/Revised: Revised November 3, 1995     

Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.     

Hyperplane arrangement cohomology and monomials in the exterior algebra

We show that if is the complement of a complex hyperplane arrangement, then the homology of has linear free resolution as a module over the exterior algebra on the first cohomology of . We study invariants of that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.

     

A Bijection Between Partially Directed Paths in the SymmetricWedge and Matchings

We give a bijection between partially directed paths in the symmetric wedge y =  ±x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of Janse van Rensburg, Prellberg, and Rechnitzer that was first discovered through the corresponding generating functions: The number of partially directed paths starting at the origin confined to the symmetric wedge y = ±x with k north steps is equal to the number of matchings on [2n] with k nestings.