Planar Triangulations Research Articles

Drawing Planar Graphs with Reduced Height

A polyline (resp., straight-line) drawing $\Gamma$ of a planar graph $G$ on a set $L_k$ of $k$ parallel lines is a planar drawing that maps each vertex of $G$ to a distinct point on $L_k$ and each edge of $G$ to a polygonal chain (resp., straight line segment) between its corresponding endpoints, where the bends lie on $L_k$. The height of $\Gamma$ is $k$, i.e., the number of lines used in the drawing. In this paper we establish new upper bounds on the height of polyline drawings of planar graphs using planar separators. Specifically, we show that every $n$-vertex planar graph with maximum degree $\Delta$, having an edge separator of size $\lambda$, admits a polyline drawing with height $4n/9+O(\lambda)$, where the previously best known bound was $2n/3$. Since $\lambda\in O(\sqrt{n\Delta})$, this implies the existence of a drawing of height at most $4n/9+o(n)$ for any planar triangulation with $\Delta \in o(n)$. For $n$-vertex planar $3$-trees, we compute straight-line drawings, with height $4n/9+O(1)$, which improves the previously best known upper bound of $n/2$. All these results can be viewed as an initial step towards compact drawings of planar triangulations via choosing a suitable embedding of the graph.

Scaling limits for the peeling process on random maps

Nous etudions la limite d’echelle du processus des volumes et des perimetres des regions explorees par un algorithme « d’epluchage » sur les cartes infinies aleatoires telles que l’UIPT (la triangulation infinie uniforme du plan) ou son analogue quadrangulaire l’UIPQ. Nos resultats s’appliquent en particulier a l’exploration des boules (pour la distance de graphe) completees et centrees a la racine de la carte. Dans ce cas, la limite d’echelle coincide avec le processus du perimetre et du volume des boules completees dans le plan brownien. Parmi les autres applications, mentionnons l’exploration des boules completees sur la carte duale et la percolation de premier passage avec poids exponentiels sur la carte duale. Ce dernier modele, equivalent au modele d’Eden sur la carte initiale, correspond a l’algorithme d’epluchage uniforme.

Plane Triangulations Without a Spanning Halin Subgraph II

A Halin graph is a plane graph constructed from a planar drawing of a tree by connecting all leaves of the tree with a cycle which passes around the boundary of the graph. The tree must have four or more vertices and no vertices of degree two. Halin graphs have many nice properties such as being Hamiltonian and remaining Hamiltonian after any single vertex deletion. In 1975, Lovasz and Plummer conjectured that every 4-connected plane triangulation contains a spanning Halin subgraph. We recently gave a negative answer to this conjecture. In this paper, we construct an infinite class of 5-connected plane triangulations without a spanning Halin subgraph. Our smallest example contains 512 vertices.

Recurrence of multiply-ended planar triangulations

In this note we show that a bounded degree planar triangulation is recurrent if and only if the set of accumulation points of some/any circle packing of it is polar (that is, planar Brownian motion avoids it with probability 1). This generalizes a theorem of He and Schramm [6] who proved it when the set of accumulation points is either empty or a Jordan curve, in which case the graph has one end. We also show that this statement holds for any straight-line embedding with angles uniformly bounded away from 0.

The height of faces of 3-polytopes

The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20 which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 10 for triangle-free 3-polytopes. In 1996, Horňak and Jendrol’ proved for arbitrarily 3-polytopes that h ≤ 23. In this paper we improve this bound to the sharp bound 20.

Boundaries of planar graphs: a unified approach

We give a new proof that the Poisson boundary of a planar graph coincides with the boundary of its square tiling and with the boundary of its circle packing, originally proven by Georgakopoulos [9] and Angel, Barlow, Gurel-Gurevich and Nachmias [2] respectively. Our proof is robust, and also allows us to identify the Poisson boundaries of graphs that are rough-isometric to planar graphs. We also prove that the boundary of the square tiling of a bounded degree plane triangulation coincides with its Martin boundary. This is done by comparing the square tiling of the triangulation with its circle packing.

Semigraph Folding Approach for Generalization of Planar Triangulation

We applied triangulation in the cycle graphs C n , n ≥3 and generalized to n – transformation, also we observed that on splicing and folding introduced by Tom Head and E. El-Kholy & co. respectively in C 2m , m ≥ 2 and it’s generalization leads to the resultant graph is P 2 ,G 0 3 , G 1 3 ,...,G n-1 3 whereas on splicing and semigraph folding introduced by S. Jeyabharathi & Co. in C 2m-1 , m ≥ 2 and its generalization leads to the resultant semigraph with and an edge and one semi edge.

Encoding Toroidal Triangulations

Poulalhon and Schaeffer introduced an elegant method to linearly encode a planar triangulation optimally. The method is based on performing a special depth-first search algorithm on a particular orientation of the triangulation: the minimal Schnyder wood. Recent progress toward generalizing Schnyder woods to higher genus enables us to generalize this method to the toroidal case. In the plane, the method leads to a bijection between planar triangulations and some particular trees. For the torus we obtain a similar bijection but with particular unicellular maps (maps with only one face).

Some results on the statistics of hull perimeters in large planar triangulations and quadrangulations

The hull perimeter at distance d in a planar map with two marked vertices at distance k from each other is the length of the closed curve separating these two vertices and lying at distance d from the first one (). We study the statistics of hull perimeters in large random planar triangulations and quadrangulations as a function of both k and d. Explicit expressions for the probability density of the hull perimeter at distance d, as well as for the joint probability density of hull perimeters at distances d1 and d2, are obtained in the limit of infinitely large k. We also consider the situation where the distance d at which the hull perimeter is measured corresponds to a finite fraction of k. The various laws that we obtain are identical for triangulations and for quadrangulations, up to a global rescaling. Our approach is based on recursion relations recently introduced by the author which determine the generating functions of so-called slices, i.e. pieces of maps with appropriate distance constraints. It is indeed shown that the map decompositions underlying these recursion relations are intimately linked to the notion of hull perimeters and provide a simple way to fully control them.

The [formula omitted]-graph coloring problem

No proof of the 4-color conjecture reveals why it is true; the goal has not been to go beyond proving the conjecture. The standard approach involves constructing an unavoidable finite set of reducible configurations to demonstrate that a minimal counterexample cannot exist. We study the 4-color problem from a different perspective. Instead of planar triangulations, we consider near-triangulations of the plane with a face of size 4; we call any such graph an a-graph. We state an a-graph coloring problem equivalent to the 4-color problem and then derive a coloring condition that a minimal a-graph counterexample must satisfy, expressing it in terms of equivalence classes under Kempe exchanges. Through a systematic search, we discover a family of a-graphs that satisfy the coloring condition, the fundamental member of which has order 12 and includes the Birkhoff diamond as a subgraph. Higher-order members include a string of Birkhoff diamonds. However, no member has an applicable parent triangulation that is internally 6-connected, a requirement for a minimal counterexample. Our research suggests strongly that the coloring and connectivity conditions for a minimal counterexample are incompatible; infinitely many a-graphs meet one condition or the other, but we find none that meets both.

The number of colorings of planar graphs with no separating triangles

A classical result of Birkhoff and Lewis implies that every planar graph with n vertices has at least 15⋅2n−1 distinct 5-vertex-colorings. Equality holds for planar triangulations with n−4 separating triangles. We show that, if a planar graph has no separating triangle, then it has at least (2+10−12)n distinct 5-vertex-colorings. A similar result holds for k-colorings for each fixed k≥5. Infinitely many planar graphs without separating triangles have less than 2.252n distinct 5-vertex-colorings. As an auxiliary result we provide a complete description of the infinite 6-regular planar triangulations.

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

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).

Low 5-stars in normal plane maps with minimum degree 5

In 1996, Jendrol’ and Madaras constructed a plane triangulation with minimum degree 5 in which the minimum vertex degree h(S5) of 5-stars is arbitrarily large. This construction has minor (5,5,5,5)-stars, that is 5-vertices with four 5-neighbors. It has been open if forbidding minor (5,5,5,5)-stars makes h(S5) finite.We prove that every normal plane map with minimum degree 5 and no minor (5,5,5,5)-stars satisfies h(S5)≤13 and construct such a map with h(S5)=12.

Well-covered triangulations: Part IV

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A plane (simple) graph in which each face is a triangle is called a (plane) triangulation. In the first of a sequence of three papers the authors proved that there are no 5-connected plane well-covered triangulations. Clearly, the only plane triangulation which is exactly 2-connected is the triangle K3. Two subsequent papers culminated in the proof that there are exactly four well-covered plane triangulations which are exactly 4-connected.It is the aim of the present paper to complete the characterization of well-covered plane triangulations by characterizing the infinite family of those well-covered triangulations of the plane which are exactly 3-connected.

Random walks on stochastic hyperbolic half planar triangulations

We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in (Angel and Ray, Ann Probab, in press). We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 213–234, 2016

Facial list colourings of plane graphs

Let G=(V,E,F) be a connected plane graph, with vertex set V, edge set E, and face set F. For X∈{V,E,F,V∪E,V∪F,E∪F,V∪E∪F}, two distinct elements of X are facially adjacent in G if they are incident elements, adjacent vertices, adjacent faces, or facially adjacent edges (edges that are consecutive on the boundary walk of a face of G). A list k-colouring is facial with respect to X if there is a list k-colouring of elements of X such that facially adjacent elements of X receive different colours. We prove that every plane graph G=(V,E,F) has a facial list 4-colouring with respect to X=E, a facial list 6-colouring with respect to X∈{V∪E,E∪F}, and a facial list 8-colouring with respect to X=V∪E∪F. For plane triangulations, each of these results is improved by one and it is tight. These results complete the theorem of Thomassen that every plane graph has a (facial) list 5-colouring with respect to X∈{V,F} and the theorem of Wang and Lih that every simple plane graph has a (facial) list 7-colouring with respect to X=V∪F.

The weight of faces in normal plane maps

The weight of a face in a 3-polytope is the degree-sum of its incident vertices, and the weight of a 3-polytope, w, is the minimum weight of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then w can be arbitrarily large, so we assume the absence of pyramidal faces in what follows.In 1940, Lebesgue proved that every quadrangulated 3-polytope has w≤21. In 1995, this bound was lowered by Avgustinovich and Borodin to 20. Recently, we improved it to the sharp bound 18.For plane triangulations without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that w≤29, which bound is sharp. Later, Borodin (1998) proved that w≤29 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 20 for triangle-free polytopes.In 1996, Horňák and Jendrol’ proved for arbitrarily polytopes that w≤32. In this paper we improve this bound to 30 and construct a polytope with w=30.

Dominating plane triangulations

In 1996, Tarjan and Matheson proved that if G is a plane triangulated disc with n vertices, γ(G)≤n/3, where γ(G) denotes the domination number of G, i.e. the cardinality of the smallest set of vertices S such that every vertex of G is either in S or adjacent to a vertex in S. Furthermore, they conjectured that the constant 1/3 could be improved to 1/4 for a sufficiently large n. Their conjecture remains unsettled. In the present paper, it is proved that if G is a Hamiltonian plane triangulation with n vertices and minimum degree at least 4, then γ(G)≤max{⌈2n/7⌉,⌊5n/16⌋}. It follows immediately that if G is a 4-connected plane triangulation with n vertices, then γ(G)≤max{⌈2n/7⌉,⌊5n/16⌋}.

Spectral Experiments+

ABSTRACTWe describe extensive computational experiments on spectral properties of random objects—random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere. We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.

Unimodular hyperbolic triangulations: circle packing and random walk

We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.