scholarly journals Every 5-connected planar triangulation is 4-ordered Hamiltonian

2015 ◽  
Vol 2 (2) ◽  
Author(s):  
Kenta Ozeki
Keyword(s):  
Planar Triangulation

Properties of Planar Triangulation and Its Application

2010 ◽  
pp. 519-526
Author(s):  
Ling Wang ◽  
Dianxuan Gong ◽  
Kaili Wang ◽  
Yuhuan Cui ◽  
Shiqiu Zheng
Keyword(s):  
Planar Triangulation

scholarly journals Semigraph Folding Approach for Generalization of Planar Triangulation

2016 ◽  
Vol 9 (48) ◽  
Author(s):  
K. Thiagarajan ◽  
J. Padmashree ◽  
Ponnammal Natarajan
Keyword(s):  
Planar Triangulation

scholarly journals Tilings with the neighborhood property

1996 ◽  
Vol 19 (1) ◽  
pp. 33-37
Author(s):  
Linda S. Fosnaugh ◽  
Earl S. Kramer
Keyword(s):  
Infinite Family ◽  
Planar Triangulation ◽  
Convex Plane

The neighborhoodN(T)of a tileTis the set of all tiles which meetTin at least one point. If for each tileTthere is a different tileT1such thatN(T)=N(T1)then we say the tiling has the neighborhood property (NEBP). Grünbaum and Shepard conjecture that it is impossible to have a monohedral tiling of the plane such that every tileThas two different tilesT1,T2withN(T)=N(T1)=N(T2). If all tiles are convex we show this conjecture is true by characterizing the convex plane tilings with NEBP. More precisely we prove that a convex plane tiling with NEBP has only triangular tiles and each tile has a 3-valent vertex. Removing 3-valent vertices and the incident edges from such a tiling yields an edge-to-edge planar triangulation. Conversely, given any edge-to-edge planar triangulation followed by insertion of a vertex and three edges that triangulate each triangle yields a convex plane tiling with NEBP. We exhibit an infinite family of nonconvex monohedral plane tilings with NEBP. We briefly discuss tilings ofR3with NEBP and exhibit a monohedral tetrahedral tiling ofR3with NEBP.

PIECEWISE CONSTANT WAVELETS ON TRIANGULATIONS OBTAINED BY 1-3 SPLITTING

2008 ◽  
Vol 06 (02) ◽  
pp. 209-222 ◽  
Author(s):  
DANIELA ROŞCA
Keyword(s):  
Wavelet Coefficient ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Piecewise Constant ◽  
Planar Triangulation ◽  
Numerical Example ◽  
Spherical Data ◽  
Refinement Process ◽  
Two Parameters

We construct piecewise constant wavelets on a bounded planar triangulation, the refinement process consisting of dividing each triangle into three triangles having the same area. Thus, the wavelets depend on two parameters linked by a certain relation. We perform a compression and try to compare different norms of the compression error, when one wavelet coefficient is canceled. Finally, we show how this construction can be moved on to the two-dimensional sphere and sphere-like surfaces, avoiding the distortions around the poles, which occur in other approaches. As numerical example, we perform a compression of some spherical data and calculate some norms of the compression error for different compression rates. The main advantage is the orthogonality and sparsity of the decomposition and reconstruction matrices.

scholarly journals Growth and percolation on the uniform infinite planar triangulation

2003 ◽  
Vol 13 (5) ◽  
pp. 935-974 ◽  
Author(s):  
Omer Angel
Keyword(s):  
Planar Triangulation ◽  
Infinite Planar

scholarly journals Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

2016 ◽  
Vol 53 (3) ◽  
pp. 846-856 ◽  
Author(s):  
Andrea Collevecchio ◽  
Abbas Mehrabian ◽  
Nick Wormald
Keyword(s):  
Related Result ◽  
Plane Graph ◽  
Time Step ◽  
Single Vertex ◽  
Planar Triangulation ◽  
The Face ◽  
Longest Paths ◽  
Recursive Trees ◽  
Positive Integers ◽  
Iterative Construction

AbstractLet r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯d,t has less than tδ vertices. The proof involves analysis that also yields a related result. 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. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t𝛿, which verifies a conjecture of Cooper and Frieze (2015).

Planar Triangulation

2016 ◽  
pp. 99-105
Author(s):  
Kenichi Kanatani ◽  
Yasuyuki Sugaya ◽  
Yasushi Kanazawa
Keyword(s):  
Planar Triangulation
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