Adjacent Triangles Research Articles

Local Condition for Planar Graphs of Maximum Degree 6 to be Total 8-Colorable

Recently Sun et al [X.-Y. Sun, J.-L. Wu, Y.-W. Wu, J.-F. Hou, Total colorings of planar graphs without adjacent triangles, {\em Discrete Math 309:202-206} (2009)] proved that planar graphs with maximum degree six and with no adjacent triangles are total $8$-colorable. This results implies that if every vertex of a planar graph of maximum degree six is missing either a $3$-cycle or a $4$-cycle, then the graph is total $8$-colorable. In this paper we strengthen that condition by showing that if every vertex of a planar graph of maximum degree six is missing some $k_v$-cycle for $k_v\in\{3,4,5,6,7,8\}$, then the graph is total $8$-colorable.

On the Direct Evaluation of Surface Integral Equation Impedance Matrix Elements Involving Point Singularities

The direct evaluation method tailored to the 4-D singular integrals over vertex adjacent triangles, arising in the first-kind and second-kind Fredholm surface integral equation formulations, is presented. A combination of singularity cancellation, reordering of the integrations, and one analytical integration results in 3-D integrals of sufficiently smooth functions, allowing a straightforward computation by standard Gaussian rules. Numerical results demonstrate that the uncertainty about the accuracy of the impedance matrix elements associated to the interaction of vertex adjacent triangles is safely removed.

On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles [Information Processing Letters 110 (24) (2010) 1084

On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles [Information Processing Letters 110 (24) (2010) 1084

RETRACTED ARTICLE: On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles

This article has been retracted: please see Elsevier Policy on Article Withdrawal ( http://www.elsevier.com/locate/withdrawalpolicy ). This article has been retracted at the request of the Editor-in-Chief due to a flaw in the proof of the theorem. The publisher apologizes for any inconvenience this may cause.

Reducing shading on GPUs using quad-fragment merging

Current GPUs perform a significant amount of redundant shading when surfaces are tessellated into small triangles. We address this inefficiency by augmenting the GPU pipeline to gather and merge rasterized fragments from adjacent triangles in a mesh. This approach has minimal impact on output image quality, is amenable to implementation in fixed-function hardware, and, when rendering pixel-sized triangles, requires only a small amount of buffering to reduce overall pipeline shading work by a factor of eight. We find that a fragment-shading pipeline with this optimization is competitive with the REYES pipeline approach of shading at micropolygon vertices and, in cases of complex occlusion, can perform up to two times less shading work.

Seamless Montage for Texturing Models

AbstractWe present an automatic method to recover high‐resolution texture over an object by mapping detailed photographs onto its surface. Such high‐resolution detail often reveals inaccuracies in geometry and registration, as well as lighting variations and surface reflections. Simple image projection results in visible seams on the surface. We minimize such seams using a global optimization that assigns compatible texture to adjacent triangles. The key idea is to search not only combinatorially over the source images, but also over a set of local image transformations that compensate for geometric misalignment. This broad search space is traversed using a discrete labeling algorithm, aided by a coarse‐to‐fine strategy. Our approach significantly improves resilience to acquisition errors, thereby allowing simple and easy creation of textured models for use in computer graphics.

A Triangle Product for Texture Stretch

Abstract A method is presented for calculating the singular values used in texture stretch analysis. Based on the side lengths of the triangles involved, the method benefits from efficiency gained when the calculation of side lengths is shared by adjacent triangles in a patch. The method employs a new tool called the triangle product. An analogous tetrahedral product is also discussed.

Bimetallic Ru–Cu tellurido complexes: controlled synthesis and electrochemical studies of copper halide–TeRu5 and Te2Ru4clusters

When [TeRu(5)(CO)(14)](2-) () was treated with 1 equiv. of CuX (X = Cl, Br, I) in THF, mono-CuX-TeRu(5) clusters [TeRu(5)(CO)(14)CuX](2-) (X = Cl, ; Br, ; I, ) were obtained. Clusters consist of an octahedral TeRu(5) core, in which one triangular Ru(3) plane is capped by a mu(3)-CuX fragment. For CuX (X = Cl, Br), the reaction of complex with 2 equiv. of CuX in THF at room temperature formed Cu(4)X(2)-linked di-TeRu(5) clusters [{TeRu(5)(CO)(14)}(2)Cu(4)X(2)](2-) (X = Cl, ; Br, ), while the same reaction in MeCN at -35 degrees C produced bis-CuX-TeRu(5) complexes [TeRu(5)(CO)(14)(CuX)(2)](2-) (X = Cl, ; Br, ). X-Ray analysis showed that displays a TeRu(5) core with two adjacent Ru(3) triangles each capped by a mu(3)-CuBr ligand while has two TeRu(5) cores that are linked by a mu(6)-Cu(4)Br(2) moiety. Clusters and underwent coupling reactions in THF to yield clusters and , and easily transformed to bis-CuX-Te(2)Ru(4) clusters [Te(2)Ru(4)(CO)(10)(CuX)(2)](2-) (X = Cl, ; X = Br, ) in MeCN. On the other hand, the reaction of with 2 equiv. of CuI in THF directly produced the bis-CuI-Te(2)Ru(4) cluster [Te(2)Ru(4)(CO)(10)(CuI)(2)](2-) (). The nature, stability, stepwise cluster transformation, and electrochemistry of these CuX-incorporated TeRu(5)- and Te(2)Ru(4)-based complexes are discussed systematically. In particular, the effects of CuX and the metal cores (TeRu(5)vs. Te(2)Ru(4)) on the resultant Te-Ru-Cu clusters are further elucidated by molecular orbital calculations at the B3LYP level of the density functional theory.

ICCD: Interactive Continuous Collision Detection between Deformable Models Using Connectivity-Based Culling

We present an interactive algorithm for continuous collision detection between deformable models. We introduce multiple techniques to improve the culling efficiency and the overall performance of continuous collision detection. First, we present a novel formulation for continuous normal cones and use these normal cones to efficiently cull large regions of the mesh as part of self-collision tests. Second, we introduce the concept of "procedural representative triangles" to remove all redundant elementary tests between nonadjacent triangles. Finally, we exploit the mesh connectivity and introduce the concept of "orphan sets" to eliminate redundant elementary tests between adjacent triangle primitives. In practice, we can reduce the number of elementary tests by two orders of magnitude. These culling techniques have been combined with bounding volume hierarchies and can result in one order of magnitude performance improvement as compared to prior collision detection algorithms for deformable models. We highlight the performance of our algorithm on several benchmarks, including cloth simulations, N-body simulations, and breaking objects.

Optimization for first order Delaunay triangulations

This paper discusses optimization of quality measures over first order Delaunay triangulations. Unlike most previous work, our measures relate to edge-adjacent or vertex-adjacent triangles instead of only to single triangles. We give efficient algorithms to optimize certain measures, including measures related to the area ratio of adjacent triangles, angle between outward normals of adjacent triangles (for polyhedral terrains), and number of convex vertices. Some other measures are shown to be NP-hard. These include maximum vertex degree, number of convex edges, and number of mixed vertices. For the latter two measures we provide, for any constant ε > 0 , factor ( 1 − ε ) approximation algorithms that run in 2 O ( 1 / ε ) ⋅ n and 2 O ( 1 / ε 2 ) ⋅ n time (when the Delaunay triangulation is given). For minimizing the maximum vertex degree, the NP-hardness proof provides an inapproximability result. Our results are presented for the class of first order Delaunay triangulations, but also apply to triangulations where for every triangle at least two edges are fixed. The approximation result on maximizing the number of convex edges is also extended to k-th order Delaunay triangulations.

Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable

The Total Coloring Conjecture, in short, TCC, says that every simple graph is ( Δ + 2 ) -totally-colorable where Δ is the maximum degree of the graph. Even for planar graphs this conjecture has not been completely settled yet. However, every planar graph with Δ ≥ 9 has been proved to be ( Δ + 1 ) -totally-colorable. In this paper, we prove that planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable.

Hyperbolic Cation Diffusion Paths in α‐RbAg4I5 Type Superionic Conductors (Z. Anorg. Allg. Chem. 2008, 634, 2156–2160)

Abstracton Page 2158:The distances Δ and Δ′ from the Ag+ positions to the three adjacent I3 triangles have to be multiplied by the lattice parameter a. The correct values are Δ = 0.086(1) × a (in Å) and Δ′ = 0.084 × a (in Å), respectively.

On $(3,1)^*$-Coloring of Plane Graphs

Given positive integers $k$ and $d$, a graph $G$ is said to be $(k,d)^*$-colorable if the vertices of $G$ can be colored with $k$ colors such that every vertex has at most $d$ neighbors receiving the same color as itself. Let ${\cal G}$ be the family of plane graphs with neither adjacent triangles nor cycles of length 5. It is proved in this paper that every graph in ${\cal G}$ is $(3,1)^*$-colorable. This result is sharp in the sense that there exist non-$(2,1)^*$-colorable plane graphs with neither triangles nor cycles of length 5. As a corollary, after removing a matching, every graph in ${\cal G}$ is 3-colorable. This provides a partial solution to a conjecture of Borodin and Raspaud [J. Combin. Theory Ser. B, 93 (2003), pp. 17-27].

Planar graphs without 5- and 7-cycles and without adjacent triangles are 3-colorable

It is known that planar graphs without cycles of length from 4 to 7 are 3-colorable [O.V. Borodin, A.N. Glebov, A. Raspaud, M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory Ser. B 93 (2005) 303–311]. We improve this result by proving that every planar graph without 5- and 7-cycles and without adjacent triangles is 3-colorable. Also, we give counterexamples to the proof of the same result in [B. Xu, On 3-colorable plane graphs without 5- and 7-cycles, J. Combin. Theory Ser. B 96 (2006) 958–963].

Automatic reconstruction of a patient-specific high-order surface representation and its application to mesh generation for CFD calculations

We describe a set of procedures for the shape reconstruction and mesh generation of unstructured high-order spatial discretization of patient-specific geometries from a series of medical images and for the simulation of flows in these meshes using a high-order hp-spectral solver. The reconstruction of the shape of the boundary is based on the interpolation of an implicit function through a set of points obtained from the segmentation of the images. This approach is favoured for its ability of smoothly interpolating between sections of different topology. The boundary of the object is initially represented as an iso-surface of an implicit function defined in terms of radial basis functions. This surface is approximated by a triangulation extracted by the method of marching cubes. The triangulation is then suitably smoothed and refined to improve its quality and permit its approximation by a quilt of bi-variate spline surface patches. Such representation is often the standard input format required for state-of-the-art mesh generators. The generation of the surface patches is based on a partition of the triangulation into Voronoi regions and dual Delaunay triangulations with an even number of triangles. The quality of the triangulation is optimized by imposing that the distortion associated with the energy of deformation by harmonic maps is minimized. Patches are obtained by merging adjacent triangles and this representation is then used to generate a mesh of linear elements using standard generation techniques. Finally, a mesh of high-order elements is generated in a bottom-up fashion by creating the additional points required for the high-order interpolation and projecting them on the edges and surfaces of the quilt of patches. The methodology is illustrated by generating meshes for a by-pass graft geometry and calculating high-order CFD solutions in these meshes.

Total colorings of planar graphs without adjacent triangles

Let G be a planar graph without adjacent 3-cycles, that is, two cycles of length 3 are not incident with a common edge. In this paper, it is proved that the total coloring conjecture is true for G ; moreover, if Δ ( G ) ≥ 9 , then the total chromatic number χ ″ ( G ) of G is Δ ( G ) + 1 . Some other related results are obtained, too.

Edge-choosability of planar graphs without adjacent triangles or without 7-cycles

A graph G is edge- L -colorable, if for a given edge assignment L = { L ( e ) : e ∈ E ( G ) } , there exists a proper edge-coloring ϕ of G such that ϕ ( e ) ∈ L ( e ) for all e ∈ E ( G ) . If G is edge- L -colorable for every edge assignment L with | L ( e ) | ≥ k for e ∈ E ( G ) , then G is said to be edge- k -choosable. In this paper, we prove that if G is a planar graph with maximum degree Δ ( G ) ≠ 5 and without adjacent 3-cycles, or with maximum degree Δ ( G ) ≠ 5 , 6 and without 7-cycles, then G is edge- ( Δ ( G ) + 1 ) -choosable.

Boolean Set Operations with Cubic Algebraic Patches

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C(1)-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation.

A Topological Glass

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with n nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions.

Magnetic domain patterns in a zigzag nanowire

Nanowires of modulated widths are potential candidate for the study of shape dependence of nanoscale magnetic properties. A study of magnetic domain arrangement in zigzag nanowires fabricated by self-assembly of nanoparticles and etching technique is presented. Thin necks in the zigzags create an anisotropy which allows domains to distribute between adjacent triangles. Magnetic force microscopy shows unique magnetization patterns for different orientations of the applied field.