Planar Triangulations Research Articles

A Lower Bound for Splines on Tetrahedral Vertex Stars

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly difficult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra---we call these closed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of $C^r$ splines on closed vertex stars of degree at least $3r+2$. We show that this formula is a lower bound on the dimension of $C^r$ splines of degree at least $(3r+2)/2$. Our proof uses apolarity and the so-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We furthermore observe that arguments of Alfeld, Schumaker, and Whiteley imply that the only splines of degree at most $(3r+1)/2$ on a generic closed vertex star are global polynomials.

Invariant embeddings of unimodular random planar graphs

Consider an ergodic unimodular random one-ended planar graph G of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By “locally finite” we mean that any bounded open set intersects finitely many embedded edges. In particular, there exist invariant embeddings in the Euclidean plane for the Uniform Infinite Planar Triangulation and for the critical Augmented Galton-Watson Tree conditioned to survive. Roughly speaking, a unimodular embedding of G is one that is jointly unimodular with G when viewed as a decoration. We show that G has a unimodular embedding in the hyperbolic plane if it is invariantly nonamenable, and it has a unimodular embedding in the Euclidean plane if and only if it is invariantly amenable. Similar claims hold for representations by tilings instead of embeddings.

Circumference of essentially 4-connected planar triangulations

A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{3}(n+4)$; moreover, this bound is sharp.

4-Connected triangulations on few lines

We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all subgraphs of such triangulations. The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.

On the cyclic coloring conjecture

A cyclic coloring of a plane graph G is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph G is its cyclic chromatic number χc(G). Let Δ∗(G) be the maximum face degree of a graph G.In this note we show that to prove the Cyclic Coloring Conjecture of Borodin from 1984, saying that every connected plane graph G has χc(G)≤⌊32Δ∗(G)⌋, it is enough to do it for subdivisions of simple 3-connected plane graphs.We have discovered four new different upper bounds on χc(G) for graphs G from this restricted family; three bounds of them are tight. As corollaries, we have shown that the conjecture holds for subdivisions of plane triangulations, simple 3-connected plane quadrangulations, and simple 3-connected plane pentagulations with an even maximum face degree, for regular subdivisions of simple 3-connected plane graphs of maximum degree at least 10, and for subdivisions of simple 3-connected plane graphs having the maximum face degree large enough in comparison with the number of vertices of their longest paths consisting only of vertices of degree two.

Monochromatic Subgraphs in Iterated Triangulations

For integers $n\ge 0$, an iterated triangulation $\mathrm{Tr}(n)$ is defined recursively as follows: $\mathrm{Tr}(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $\mathrm{Tr}(n)$ is the plane triangulation obtained from the plane triangulation $\mathrm{Tr}(n-1)$ by, for each inner face $F$ of $\mathrm{Tr}(n-1)$, adding inside $F$ a new vertex and three edges joining this new vertex to the three vertices incident with $F$.
 In this paper, we show that there exists a 2-edge-coloring of $\mathrm{Tr}(n)$ such that $\mathrm{Tr}(n)$ contains no monochromatic copy of the cycle $C_k$ for any $k\ge 5$. As a consequence, the answer to one of two questions asked by Axenovich et al. is negative. We also determine the radius 2 graphs $H$ for which there exists $n$ such that every 2-edge-coloring of $\mathrm{Tr}(n)$ contains a monochromatic copy of $H$, extending a result of Axenovich et al. for radius 2 trees.

Local convergence of large random triangulations coupled with an Ising model

We prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size n n decorated by an Ising configuration with a weight proportional to the energy of this configuration. To do so, we establish the algebraicity and the asymptotic behaviour of the partition functions of triangulations with spins for any boundary condition. In particular, we show that these partition functions all have the same phase transition at the same critical temperature. Some properties of the limiting object – called the Infinite Ising Planar Triangulation – are derived, including the recurrence of the simple random walk at the critical temperature.

Total domination in plane triangulations

A total dominating set of a graph G=(V,E) is a subset D of V such that every vertex in V is adjacent to at least one vertex in D. The total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set of G. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that γt(G)≤⌊2n5⌋ for any near-triangulation G of order n≥5, with two exceptions.

Optical Dual Laser Based Sensor Denoising for OnlineMetal Sheet Flatness Measurement Using Hermite Interpolation.

Flatness sensors are required for quality control of metal sheets obtained from steel coils by roller leveling and cutting systems. This article presents an innovative system for real-time robust surface estimation of flattened metal sheets composed of two line lasers and a conventional 2D camera. Laser plane triangulation is used for surface height retrieval along virtual surface fibers. The dual laser allows instantaneous robust and quick estimation of the fiber height derivatives. Hermite cubic interpolation along the fibers allows real-time surface estimation and high frequency noise removal. Noise sources are the vibrations induced in the sheet by its movements during the process and some mechanical events, such as cutting into separate pieces. The system is validated on synthetic surfaces that simulate the most critical noise sources and on real data obtained from the installation of the sensor in an actual steel mill. In the comparison with conventional filtering methods, we achieve at least a 41% of improvement in the accuracy of the surface reconstruction.

Hamiltonian cycles in 4-connected plane triangulations with few 4-separators

Hakimi, Schmeichel and Thomassen showed in 1979 that every 4-connected triangulation on n vertices has at least n∕log2n hamiltonian cycles, and conjectured that the sharp lower bound is 2(n−2)(n−4). Recently, Brinkmann, Souffriau and Van Cleemput gave an improved lower bound 125(n−2). In this paper we show that every 4-connected triangulation with O(logn) 4-separators has Ω(n2∕log2n) hamiltonian cycles.

The last temptation of William T. Tutte

In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times of the Four Color Conjecture. He asked whether the 4-coloring complex of a planar triangulation could have two components in which all colorings had the same parity. In this note we answer Tutte’s question contrary to his speculations by showing that there are triangulations of the plane whose coloring complexes have arbitrarily many even and odd components.

Anomalous diffusion of random walk on random planar maps

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most n^{1/4 + o_n(1)} in n units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after n steps is n^{1/4 + o_n(1)}, as conjectured by Benjamini and Curien (Geom Funct Anal 2(2):501–531, 2013. arXiv:1202.5454). More generally, we show that the simple random walks on a certain family of random planar maps in the gamma -Liouville quantum gravity (LQG) universality class for gamma in (0,2)—including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps—typically travels graph distance n^{1/d_gamma + o_n(1)} in n units of time, where d_gamma is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on gamma by Ding and Gwynne (Commun Math Phys 374:1877–1934, 2018. arXiv:1807.01072). Since d_gamma > 2, this shows that the simple random walk on each of these maps is subdiffusive. Our proofs are based on an embedding of the random planar maps under consideration into {mathbb {C}} wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG.

Local limits of uniform triangulations in high genus

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the planar stochastic hyperbolic triangulations (PSHT) defined in Curien (Probab Theory Relat Fields 165(3):509–540, 2016). The proof relies on a combinatorial argument and the Goulden–Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane T is weakly Markovian in the sense that the probability to observe a finite triangulation t around the root only depends on the perimeter and volume of t, then T is a mixture of PSHT.

Site percolation and isoperimetric inequalities for plane graphs

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini, and Horesh that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.

On Spanning Trees without Vertices of Degree 2 in Plane Triangulations

On Spanning Trees without Vertices of Degree 2 in Plane Triangulations

Restricted matching in plane triangulations and near triangulations

We present a complete description of matching extendability in 5-connected planar triangulations after the deletion of a set of vertices of appropriate parity. We also introduce a variation of distance restricted matching extension in the setting of vertex deleted graphs and prove some results which generalize certain previous results obtained for 5-connected planar triangulations.

A mating-of-trees approach for graph distances in random planar maps

We introduce a general technique for proving estimates for certain random planar maps which belong to the gamma -Liouville quantum gravity (LQG) universality class for gamma in (0,2). The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; gamma =sqrt{8/3}); and planar maps weighted by the number of different spanning trees (gamma =sqrt{2}), bipolar orientations (gamma =sqrt{4/3}), or Schnyder woods (gamma =1) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of gamma -LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when gamma =sqrt{8/3}, we instead deduce estimates for the sqrt{8/3}-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

Distance restricted matching extension missing vertices and edges in 5‐connected triangulations of the plane

AbstractIn [4] it was shown that in a 5‐connected even planar triangulation G, every matching M of size can be extended to a perfect matching of G, as long as the edges of M lie at distance at least 5 from each other. Somewhat later in [7], the following result was proved. Let be a 5‐connected triangulation of a surface different from the sphere. Let be the Euler characteristic of . Suppose with even and and are two matchings in such that . Further suppose that the pairwise distance between two elements of is at least 5 and the face‐width of the embedding of in is at least . Then there is a perfect matching in which contains such that . In the present paper, we present some results which, in a sense, lie in the gap between the two above theorems, in that they deal with restricted matching extension in a planar triangulation when a set of vertices which lie pairwise at sufficient distance from one another has been deleted. In particular, we prove a planar analogue of the result in [7] stated above.

Conformally Symmetric Triangular Lattices and Discrete ϑ-Conformal Maps

Abstract Two immersed triangulations in the plane with the same combinatorics are considered as preimage and image of a discrete immersion $F$. We compare the cross-ratios $Q$ and $q$ of corresponding pairs of adjacent triangles in the two triangulations. If for every pair the arguments of these cross-ratios (i.e., intersection angles of circumcircles) agree, $F$ is a discrete conformal map based on circle patterns. Similarly, if for every pair the absolute values of the corresponding cross-ratios $Q$ and $q$ (i.e., length cross-ratios) agree, the two triangulations are discretely conformally equivalent. We introduce a new notion, discrete $\vartheta $-conformal maps, which interpolates between these two known notions of discrete conformality for planar triangulations. We prove that there exists an associated variational principle. In particular, discrete $\vartheta $-conformal maps are unique maximizers of a locally defined concave functional ${\mathcal{F}}_{\vartheta}$in suitable variables. Furthermore, we study conformally symmetric triangular lattices that contain examples of discrete $\vartheta $-conformal maps.

Hamiltonian-connectedness of triangulations with few separating triangles

We prove that 3-connected plane triangulations containing a single edge contained in all separating triangles are hamiltonian-connected. As a direct corollary we have that 3-connected plane triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any s >= 4 there are 3-connected triangulation with s separating triangles that are not hamiltonian-connected. We also present computational results which show that all `small' 3-connected triangulations with at most 3 separating triangles are hamiltonian-connected.