On Hardness of the Joint Crossing Number

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar

A History of Flips in Combinatorial Triangulations

Given two combinatorial triangulations, how many edge flips are necessary and sufficient to convert one into the other? This question has occupied researchers for over 75 years. We provide a comprehensive survey, including full proofs, of the various attempts to answer it.Comment: Added a paragraph referencing earlier work in the vertex-labelled setting that has implications for the unlabeled settin

Ergodicity and Slowing Down in Glass-Forming Systems with Soft Potentials: No Finite-Temperature Singularities

The aim of this paper is to discuss some basic notions regarding generic glass forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction we discuss the so called `glass transition' in which super-cooled amorphous state is formed, accompanied with a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of N particles with different interaction length-scales we show that (i) the system remains ergodic at all temperatures. (ii) the number of topologically distinct configurations can be computed, is temperature independent, and is exponential in N. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in N, showing that the graph of configurations has the `small world' property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the super-Arrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0Comment: 10 pages, 9 figures, submitted to PR

Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.Comment: 25 pages, 10 figure

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor improvements. 44 pages, 14 figure

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number

Itsy bitsy topological field theory

We construct an elementary, combinatorial kind of topological quantum field theory, based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda--Kazez--Matic. This topological field theory stores information in binary format on a surface and has "digital" creation and annihilation operators, giving a toy-model embodiment of "it from bit".Comment: 54 pages, 35 figures. Minor edits, extra figures adde

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus.Comment: 14 pages, 11 figures, only minor changes from first version, to appear in Geometriae Dedicat

Advertising media strategies in the film industry

The primary aim of this article is to estimate the multiple determinants of film advertising expenditures in four important media, namely television, press, outdoor and radio, in the UK. First, television advertising, the leading film advertising medium, is examined as part of a system of equations, capturing the interdependences between advertising, the number of screens on which films are initially shown and box office revenues. Then a reduced form model is put forward to reveal the determinants of film advertising in the four media. While major distribution companies have different preferences for the use of the alternative advertising media, results highlight the importance of quality signals, such as critical reviews, in determining advertising expenditures in the film industry. Moreover, advertising expenditures can themselves be considered to offer potential cinema goers signals of film quality