Coloring decompositions of complete geometric graphs

A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erd\H{o}s-Faber-Lov\'asz states that every decomposition $[K_n,P]$ of the complete graph $K_n$ satisfies $\chi'([K_n,P])\leq n$. In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case in which the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure

Searching edges in the overlap of two plane graphs

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains one of which is convex in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n) time and O(n+m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n axis-aligned rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure

Coloring half-planes and bottomless rectangles

We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms

Statistical properties of filaments in weak gravitational lensing

We study weak lensing properties of filaments that connect clusters of galaxies through large cosmological $N$-body simulations. We select 4639 halo pairs with masses higher than $10^{14}h^{-1}\mathrm{M}_\odot$ from the simulations and investigate dark matter distributions between two haloes with ray-tracing simulations. In order to classify filament candidates, we estimate convergence profiles and perform profile fitting. We find that matter distributions between haloes can be classified in a plane of fitting parameters, which allow us to select straight filaments from the ray-tracing simulations. We also investigate statistical properties of these filaments, finding them to be consistent with previous studies. We find that $35\%$ of halo pairs possess straight filaments, $4\%$ of which can directly be detected at $S/N\geq2$ with weak lensing. Furthermore, we study statistical properties of haloes at the edges of filaments. We find that haloes are preferentially elongated along filamentary structures and are less massive with increasing filament masses. However, the dependence of these halo properties on masses of straight filaments is very weak.Comment: 14 pages, 12 figures, 3 tables. Accepted for publication in MNRA

STM observation of electronic wave interference effect in finite-sized graphite with dislocation-network structures

Superperiodic patterns near a step edge were observed by STM on several-layer-thick graphite sheets on a highly oriented pyrolitic graphite substrate, where a dislocation network is generated at the interface between the graphite overlayer and the substrate. Triangular- and rhombic-shaped periodic patterns whose periodicities are around 100 nm were observed on the upper terrace near the step edge. In contrast, only outlines of the patterns similar to those on the upper terrace were observed on the lower terrace. On the upper terrace, their geometrical patterns gradually disappeared and became similar to those on the lower terrace without any changes of their periodicity in increasing a bias voltage. By assuming a periodic scattering potential at the interface due to dislocations, the varying corrugation amplitudes of the patterns can be understood as changes in LDOS as a result of the beat of perturbed and unperturbed waves, i.e. the interference in an overlayer. The observed changes in the image depending on an overlayer height and a bias voltage can be explained by the electronic wave interference in the ultra-thin overlayer distorted under the influence of dislocation-network structures.Comment: 8 pages; 6 figures; Paper which a part of cond-mat/0311068 is disscussed in detai

Morsifications and mutations

We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.Comment: Major revision and expansion. Several new results. 80 pages, 66 figure