Planar Triangulations Research Articles

Non-degenerated Ground States and Low-degenerated Excited States in the Antiferromagnetic Ising Model on Triangulations

We study the unexpected asymptotic behavior of the degeneracy of the first few energy levels in the antiferromagnetic Ising model on triangulations of closed Riemann surfaces. There are strong mathematical and physical reasons to expect that the number of ground states (i.e., degeneracy) of the antiferromagnetic Ising model on the triangulations of a fixed closed Riemann surface is exponential in the number of vertices. In the set of plane triangulations, the degeneracy equals the number of perfect matchings of the geometric duals, and thus it is exponential by a recent result of Chudnovsky and Seymour. From the physics point of view, antiferromagnetic triangulations are geometrically frustrated systems, and in such systems exponential degeneracy is predicted. We present results that contradict these predictions. We prove that for each closed Riemann surface S of positive genus, there are sequences of triangulations of S with exactly one ground state. One possible explanation of this phenomenon is that exponential degeneracy would be found in the excited states with energy close to the ground state energy. However, as our second result, we show the existence of a sequence of triangulations \({(\mathcal{T}_n)}\) of a closed Riemann surface of genus 10 with exactly one ground state such that the degeneracy of each of the 1st, 2nd, 3rd and 4th excited energy levels belongs to O(n), O(n2), O(n3) and O(n4), respectively.

Every 3-polytope with minimum degree 5 has a 6-cycle with maximum degree at most 11

Let φP(C6) (respectively, φT(C6)) be the minimum integer k with the property that every 3-polytope (respectively, every plane triangulation) with minimum degree 5 has a 6-cycle with all vertices of degree at most k. In 1999, S. Jendrol’ and T. Madaras proved that 10≤φT(C6)≤11. It is also known, due to B. Mohar, R. Škrekovski and H.-J. Voss (2003), that φP(C6)≤107.We prove that φP(C6)=φT(C6)=11.

Small degeneracy of antiferromagnetic triangulations

We study the unexpected asymptotic behavior of the degeneracy of the first few energy levels in the antiferromagnetic Ising model on triangulations of closed Riemann surfaces.There are strong mathematical and physical reasons to expect that the number of ground states (i.e., ground state degeneracy) of antiferromagnetic triangulations is exponential in the number of vertices. In the set of plane triangulations, this degeneracy equals the number of perfect matchings in the geometric dual, and thus it is exponential by recent result of Chudnovsky and Seymour. From the physics point of view, antiferromagnetic triangulations are geometrically frustrated systems, and in such systems exponential degeneracy is predicted. We present results that contradict these predictions.We show that for each closed Riemann surface Ω of positive genus, there are sequences of triangulations of Ω with exactly one ground state. One possible explanation of this phenomenon is that exponential degeneracy would be found in the excited states with energy close to the energy of the ground state. However, as our second result, we present a sequence of triangulations (Tn) of a closed Riemann surface of genus 10 with exactly one ground state, and such that the degeneracy of the first four energy levels is a polynomial in n, where n is the number of vertices of Tn.

Fingerprint verification using characteristic vectors based on planar graphics

This paper describes a new characteristic vector model for fingerprint representation that uses a planar graph and triangulation algorithms. This new characteristic vector model performed better in a fingerprint identification system than other vector models already proposed in the literature. Minutiae extraction is an essential step in a fingerprint recognition system. This paper presents a new method for minutiae extraction that explores the duality ridge ending/ridge bifurcation that exists when the skeleton image of a fingerprint is inverted. This new extraction method simplifies the computational complexity of a fingerprint identification system.

On the Longest Paths and the Diameter in Random Apollonian Networks

Abstract We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n − 3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) every path in a RAN has length o ( n ) , refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a path of length ( 2 n − 5 ) log 2 / log 3 , and that the expected length of its longest path is Ω ( n 0.88 ) . Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to c log n , where c ≈ 1.668 is the solution of an explicit equation.

On the Riemann surface type of random planar maps

We show that the (random) Riemann surfaces of the Angel–Schramm uniform infinite planar triangulation and of Sheffield’s infinite necklace construction are both parabolic. In other words, Brownian motion on these surfaces is recurrent. We obtain this result as a corollary to a more general theorem on subsequential distributional limits of random unbiased disc triangulations, following work of Benjamini and Schramm.

Planar maps, circle patterns and 2D gravity

Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point process. We show that this measure can be expressed as: (1) a sum over 3-spanning-trees partitions of the edges of the Delaunay triangulations; (2) the volume form of a Kähler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space \mathbb{H}_3 ; (3) a discretized version (involving finite difference complex derivative operators \nabla,\bar\nabla ) of Polyakov's conformal Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes, thus also establishing a link with topological 2d gravity.

Recurrence of planar graph limits

We prove that any distributional limit of nite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform innite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm. We also settle another related problem of Benjamini and Schramm. We show that in any bounded degree, nite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C logT .

A simple routing algorithm based on Schnyder coordinates

Geometric routing is an elegant way for solving network routing problems. The essence of this routing scheme is the following: When an origin vertex u wants to send a message to a destination vertex w, it forwards the message to a neighbor t, solely based on the location information of u,w and the neighbors of u. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. A greedy drawing of a graph G is an embedding of G for which the greedy routing algorithm works. Recently, Leighton and Moitra (2010) [18] found an algorithm that produces an embedding of any 3-connected planar graph in R2 that supports greedy routing. A similar result was independently found by Angelini et al. (2010) [2]. One main drawback of these algorithms is that they need Ω(nlogn) bits to represent the coordinates of the vertex locations. This is the same space usage as traditional routing table approaches, and hence makes greedy routing infeasible in applications.In the greedy routing scheme, the routing decision is based on decreasing distances between vertex locations. For this idea to work, however, the routing decision does not have to be based on decreasing distances. As long as the routing decision is solely based on the location information of the source, the destination, and the neighbors of the source, the geometric routing scheme will work fine.In this paper, we introduce a new model of geometric routing. Instead of relying on decreasing distance for routing decisions, our algorithm uses other criterion to determine the routing path, solely based on location information. Our routing algorithms are based on Schnyder coordinates which are derived from Schnyder realizers for plane triangulations and Schnyder woods for 3-connected plane graphs. The coordinates of vertex locations consist of three integers between 0 and 2n, hence the representation only needs O(logn) bits. In order to send a message from the origin u to the destination w, our routing algorithm determines the routing path from the Schnyder coordinates of u, w and all neighbors of u. The algorithms are natural and simple to implement.

Phase transitions inXYantiferromagnets on plane triangulations

Using Monte Carlo simulations and finite-size scaling, we investigate the $XY$ antiferromagnet on the triangular, Union Jack, and bisected-hexagonal lattices, and in each case find both Ising and Kosterlitz-Thouless transitions. As is well known, on the triangular lattice, as the temperature decreases the system develops chiral order for temperatures $T<{T}_{\text{c}}$, and then quasi-long-range magnetic order on its sublattices when $T<{T}_{\text{s}}$, with ${T}_{\text{s}}<{T}_{\text{c}}$. On the Union Jack and bisected-hexagonal lattices, by contrast, we find that as $T$ decreases the magnetizations on some of the sublattices become quasi-long-range ordered at a temperature ${T}_{\text{s}}>{T}_{\text{c}}$, before chiral order develops. In some cases, the sublattice spins then undergo a second transition, of Ising type, separating two quasi-long-range ordered phases. On the Union Jack lattice, the magnetization on the degree-4 sublattice remains disordered until ${T}_{\text{c}}$ and then undergoes an Ising transition to a quasi-long-range ordered phase.

On the Page Number of Upward Planar Directed Acyclic Graphs

In this paper we study the page number of upward planar directed acyclic graphs. We prove that: (1) the page number of any n-vertex upward planar triangulation G whose every maximal 4-connected component has page number k is at most min {O(klogn),O(2k)}; (2) every upward planar triangulation G with $o(\frac{n}{\log n})$ diameter has o(n) page number; and (3) every upward planar triangulation has a vertex ordering with o(n) page number if and only if every upward planar triangulation whose maximum degree is $O(\sqrt n)$ does.

On Succinct Greedy Drawings of Plane Triangulations and 3-Connected Plane Graphs

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in the Euclidean plane R2 (by Papadimitriou and Ratajczak in Theor. Comp. Sci. 344(1):3–14, 2005). Leighton and Moitra (Discrete Comput. Geom. 44(3):686–705, 2010) recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations require Ω(nlogn) bits to represent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications.

A STATIC OPTIMALITY TRANSFORMATION WITH APPLICATIONS TO PLANAR POINT LOCATION

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2-D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.

Construction of Generalized Ricci Flow Based Virtual Coordinates for Wireless Sensors Network

Geographic routing algorithms are well-suited to the special characteristics of WSNs. However, the performance of geographic routing is impacted by physical voids and localization errors. In this paper, we use conformal mapping and Möbius transformation to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we perform generalized Ricci flow to a planar triangulation from the sensor network topology with non-triangular faces as holes such that all the non-triangular faces are mapped to perfect circles. The generalized Ricci flow is more flexible, more robust and conformal for meshes with low quality triangulations. The generalized method allows the two circles to either intersect or separate from each other. This greatly improves the flexibility and robustness of the method. In addition, we transform the planar triangulation with perfect circles into a circular domain D in that the probability of obtuse triangles is greatly decreased. Thus greedy forwarding will never get stuck at an intermediate node. Simulation results are presented for these scenarios.

The structure of chromatic polynomials of planar triangulations and implications for chromatic zeros and asymptotic limiting quantities

We present an analysis of the structure and properties of chromatic polynomials of one-parameter and multi-parameter families of planar triangulation graphs , where is a vector of integer parameters. We use these to study the ratio of to the Tutte upper bound (τ − 1)n − 5, where and n is the number of vertices in . In particular, we calculate limiting values of this ratio as n → ∞ for various families of planar triangulations. We also use our calculations to analyze zeros of these chromatic polynomials. We study a large class of families with p = 1 and p = 2 and show that these have a structure of the form for p = 1, where λ1 = q − 2, λ2 = q − 3, and λ3 = −1, and for p = 2. We derive properties of the coefficients and show that has a real chromatic zero that approaches as one or more of the mi → ∞. The generalization to p ⩾ 3 is given. Further, we present a one-parameter family of planar triangulations with real zeros that approach 3 from below as m → ∞. Implications for the ground-state entropy of the Potts antiferromagnet are discussed.

Succinct geometric indexes supporting point location queries

We propose designing data structures called succinct geometric indexes of negligible space (more precisely, o ( n ) bits) that support geometric queries in optimal time, by taking advantage of the n points in the dataset permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O (lg n ) time. We also design three variants of this index. The first supports point location using lg n + 2√lg n + O (lg 1/4 n ) point-line comparisons. The second supports point location in o (lg n ) time when the coordinates are integers bounded by U . The last variant can answer point location queries in O ( H + 1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O ( n ) words or O(n lg n ) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O (lg 2 n ) time.

Chromatic polynomials of planar triangulations, the Tutte upper bound and chromatic zeros

Tutte proved that if Gpt is a planar triangulation and P(Gpt, q) is its chromatic polynomial, then |P(Gpt, τ + 1)| ⩽ (τ − 1)n − 5, where and n is the number of vertices in Gpt. Here we study the ratio r(Gpt) = |P(Gpt, τ + 1)|/(τ − 1)n − 5 for a variety of planar triangulations. We construct infinite recursive families of planar triangulations Gpt, m depending on a parameter m linearly related to n and show that if P(Gpt, m, q) only involves a single power of a polynomial, then r(Gpt, m) approaches zero exponentially fast as n → ∞. We also construct infinite recursive families for which P(Gpt, m, q) is a sum of powers of certain functions and show that for these, r(Gpt, m) may approach a finite nonzero constant as n → ∞. The connection between the Tutte upper bound and the observed chromatic zero(s) near to τ + 1 is investigated. We report the first known graph for which the zero(s) closest to τ + 1 is not real, but instead is a complex-conjugate pair. Finally, we discuss connections with the nonzero ground-state entropy of the Potts antiferromagnet on these families of graphs.

Proximity thresholds for matching extension in the torus and Klein bottle

A graph G is said to have the property Ed(m,n) if, given any two disjoint matchings M and N such that the edges within M are pair-wise distance at least d from each other as are the edges in N, there is a perfect matching F in G such that M⊆F and F∩N=0̸. This property has been previously studied for planar triangulations as well as projective planar triangulations. Here this study is extended to triangulations of the torus and Klein bottle.

Mixing times of Markov chains on 3-Orientations of Planar Triangulations

Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a $\textit{Schnyder wood}$ that has proven useful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing'' chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. We show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping'' chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing.

Polygon Morphing and Its Application in Orebody Modeling

Three different polygon morphing methods are examined. The first one is based on the utilization of the trimmed skeleton of the symmetric difference of the source and target polygons as an intermediate polygon. The second one reduces the problem to the problem of morphing compatible planar triangulations and utilizes the representation of planar triangulations as a matrix constructed using barycentric coordinates of the planar triangulation′s vertices relative to their neighbors. The third and last one describes the polygon by the parametric curve representation based on estimated Fourier parameters and thus transfers the morphing process to Fourier parametric space. The different features and comparative results of these methods are shown by the tests with different examples. These methods are used for generating a set of polygonal sections from two nonplanar polygonal sections which are nearly planar in 3D before constructing a three‐dimensional object from these nonplanar sections.