On groups generated by two positive multi-twists: Teichmueller curves and Lehmer's number

From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these determine the same set of Teichmueller curves as the non-obtuse lattice triangles which were classified by Kenyon, Smillie, and Puchta. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under consideration. In addition, we describe a connection to work of McMullen on Coxeter groups and related work of Hironaka on a construction of an interesting class of fibered links.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper36.abs.htm

Coning-off CAT(0) cube complexes

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical hyperbolicity of $C'(1/4)-T(4)$ small cancellation quotients of free products.Comment: 45 pages, 13 figures. Comments are welcom

Triangle-free geometric intersection graphs with large chromatic number

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect and $\chi(\mathcal{F})>k$. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries, and equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.Comment: Small corrections, bibliography updat

Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Witness (Delaunay) Graphs

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable generalizations. We introduce a generalization, witness graphs, that encompasses both the goal of more power and flexibility for graph drawing issues and a wider spectrum for neighborhood analysis. We study in detail two concrete examples, both related to Delaunay graphs, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200