Planar Triangulations Research Articles

On sorting triangles in a delaunay tessellation

In a two-dimensional Delaunay-triangulated domain, there exists a partial ordering of the triangles (with respect to a vertex) that is consistent with the two-dimensional visibility of the triangles from that vertex. An equivalent statement is that a polygon that is star-shaped with respect to a given vertex can be extended, one triangle at a time, until it includes the entire domain. Arbitrary planar triangulations do not possess this useful property which allows incremental processing of the triangles.

Diagonal 11‐coloring of plane triangulations

AbstractThe vertices of each plane triangulation without loops and multiple edges may be colored with 11 colors so that for every two adjacent triangles [xyz] and [wxy], the vertices x,y,w,z are colored pairwise differently.

On chromatic and dichromatic sum equations

This paper provides firstly an exposition on the theory of chromatic sums for rooted planar triangulations founded by W.T. Tutte. Then, the generalization of the theory for rooted nonseparable planar maps is described. Further, a functional equation of the dichromatic sum functions for rooted nonseparable planar maps is also established by using the same decomposition lemmas as those in the chromatic case. However, it seems that the equations obtained here can not be directly derived from the dichromatic sum equation for rooted general planar maps which was discovered by W.T. Tutte as well.

Connectivity of plane triangulations

This paper gives a topological characterization of the sets of articulation in planar graphs. It leads to linear algorithms for testing the 3-connectivity, 4-connectivity and 5-connectivity for plane triangulations (i.e., topological planar graphs such that all faces, except possibly the external face, are circuits of length 3). These algorithms remain optimal when they are extended in order to enumerate all the articulation k-sets of a k-connected triangulation. This study uses subgraph listing algorithms developed by Chiba and Nishizeki. It is related to the Hamiltonian circuit problem: since all 4-connected planar graphs are Hamiltonian, and there are linear algorithms for finding Hamiltonian circuits in such graphs, the 4-connectivity test means that there is a 2-step linear process for finding Hamiltonian circuits that is guaranteed to work for 4-connected plane triangulations.

Hamiltonian cycles in planar triangulations with no separating triangles

AbstractA classical theorem of Hassler Whitney asserts that any maximal planar graph with no separating triangles is Hamiltonian. In this paper, we examine the problem of generalizing Whitney's theorem by relaxing the requirement that the triangulation be a maximal planar graph (i.e., that its outer boundary be a triangle) while maintaining the hypothesis that the triangulation have no separating triangles. It is shown that the conclusion of Whitney's theorem still holds if the chords satisfy a certain sparse‐ness condition and that a Hamiltonian cycle through a graph satisfying this condition can be found in linear time. Upper bounds on the shortness coefficient of triangulations without separating triangles are established. Several examples are given to show that the theorems presented here cannot be extended without strong additional hypotheses. In particular, a 1‐tough, non‐Hamiltonian triangulation with no separating triangles is presented.

On the penrose number of cubic diagrams

A cubic diagram is a cubic graph G drawn in the plane, possibly with edge-crossings. The drawing defines a sign for each edge-3-coloring of G. The Penrose number of G is the sum of signs of its edge-3-colorings. For plane graphs it coincides with the number of edge-3-colorings. Given a cubic diagram G, we define a sign for every Eulerian orientation of its line-graph L( G) and prove that the Penrose number of G is equal to the sumof signs of the Eulerian orientations of L( G). This yields a new recursive scheme for the computation of the Penrose number. Another consequence is a simple formula which gives the number of vertex-4- colorings of a loopless plane triangulation in terms of the mappings from the vertex-set to {1,2,3} which take exactly two distinct values on each triangle.

On the number of Hamiltonian cycles in triangulations

AbstractIt is proved that if a planar triangulation different from K3 and K4 contains a Hamiltonian cycle, then it contains at least four of them. Together with the result of Hakimi, Schmeichel, and Thomassen [2], this yields that, for n ⩾ 12, the minimum number of Hamiltonian cycles in a Hamiltonian planar triangulation on n vertices is four. We also show that this theorem holds for triangulations of arbitrary surfaces and for 3‐connected triangulated graphs.

A separation property of planar triangulations

AbstractEvery planar triangulation G has the property that each induced cycle C of length at least 4 in G separates G, but no proper subgraph of C does. This property is trivially shared by all chordal graphs since these contain no such cycles at all. We ask to what extent maximally planar graphs and chordal graphs are unique with this property — or how much larger the class of graphs is that it determines. The answer is given in the form of a characterization of this class in terms of the simplicial decompositions of its elements. The theory of simplicial decompositions appears to be a very interesting, but still largely unexploited, method of characterization in graph theory, which seems tailor‐made for problems like the one discussed.

Interval representations of planar graphs

We prove that every planar graph is the intersection graph of a collection of three-dimensional boxes, with intersections occuring only in the boundaries of the boxes. Furthermore, we characterize the graphs that have such representations (called strict representations) in the plane. These are precisely the proper subgraphs of 4-connected planar triangulations, which we characterize by forbidden sub-graphs. Finally, we strengthen a result of E. R. Scheinerman (“Intersection Classes and Multiple Intersection Parameters”, Ph. D. thesis, Princeton Univ., 1984) to show that every planar graph has a strict representation using at most two rectangles per vertex.

Almost all Convex Polyhedra are Asymmetric

Although many types of rooted planar maps have been enumerated (see [8] for example), not much has been done on enumeration of entirely unrooted planar maps. Yet in virtually all cases of interest, it has appeared that comparatively very few of the maps are symmetric (have non-trivial automorphisms). This suggests that an asymptotic formula for the numbers of unrooted maps of a particular type on n edges can be obtained by dividing the numbers of rooted maps of that type on n edges by 4n, where 4n is the number of potentially distinct rootings of an asymmetric n-edged map. The assertion that almost all maps of a given type are asymmetric has previously been proved in only two non-trivial cases: for 3-connected planar triangulations by Tutte [9] and for all n-edged 3-connected planar maps in [5]. We prove here that it is also true for 3-connected planar maps with a given number of vertices and faces, uniformly as either parameter approaches infinity.

Rectangular duals of planar graphs

AbstractLet one face of a cube be dissected into rectangles, no 4 of which meet at a single point. The dual graph of this configuration is a 4‐connected triangulated plane graph. This paper shows that any 4‐connected plane triangulation with at least 6 vertices and at least one vertex of degree 4 is dual to a cube with one face dissected into rectangles. The proof of this result contains an implicit algorithm for obtaining such a dissection. The paper also discusses a related problem: Given a graph G with all faces triangular except the outer face, does there exist a dissection of a rectangle into rectangles for which G describes the adjacency relations among the rectangles?

Computational Geometry on a Systolic Chip

This paper describes systolic algorithms for a number of geometric problems. For the sake of realism we restrict our investigation to one-dimensional arrays whose communication links with the outside are located at the end cells. Implementations yielding maximal throughput are given for solving dynamic versions of convex hull, inclusion, range and intersection search, planar point location, intersection, triangulation, and closest-point problems.

A generalization of chordal graphs

AbstractIn a 3‐connected planar triangulation, every circuit of length ≥ 4 divides the rest of the edges into two nontrivial parts (inside and outside) which are “separated” by the circuit. Neil Robertson asked to what extent triangulations are characterized by this property, and conjectured an answer. In this paper we prove his conjecture, that if G is simple and 3‐connected and every circuit of length ≥ 4 has at least two “bridges,” then G may be built up by “clique‐sums” starting from complete graphs and planar triangulations. This is a generalization of Dirac's theorem about chordal graphs.

Chromatic Solutions, II

This paper is a continuation of the Waterloo Research Report CORR 81-12, (see [1]) referred to in what follows as I. That Report is entitled “Chromatic Solutions”. It is largely concerned with a power series h in a variable z2, in which the coefficients are polynomials in a “colour number” λ. By definition the coefficient of z2r, where r > 0, is the sum of the chromatic polynomials of the rooted planar triangulations of 2r faces. (Multiple joins are allowed in these triangulations.) Thus for a positive integral λ the coefficient is the number of λ-coloured rooted planar triangulations of 2r faces. The use of the symbol z2 instead of a simple letter t is for the sake of continuity with earlier papers.In I we consider the case(1)where n is an integer exceeding 4.

An unavoidable set of configurations in planar triangulations

Let T be a normal triangulation (considered in the context of the four-color problem). Assume that no two vertices of degree 5 are adjacent. Then T contains at least 1 of the 47 configurations in Table I all of which are likely to be 4-color reducible.

On a pair of functional equations of combinatorial interest

The equations of the title appear in the author's paper “Chromatic Sums for Rooted Planar Triangulations, V: Special Equations.” (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy2 of two variablesy1 andz. A parameterμ is involved. The main result is that forμ = 2 cos (2π/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.

Chromatic Sums for Rooted Planar Triangulations, V: Special Equations

In I we obtained an equation, called the "chromatic equation", for the generating function g(x, y, z, λ). In II and III we obtained special equations, valid in the cases λ = τ + 1 and λ = 3 respectively, for the generating function l(y, z, λ), defined as the coefficient of x2 in g(x, y, z, λ). The argument was independent of that in I and no attempt was made to derive the new formulae from the chromatic equation.

A census of simple planar triangulations

Explicit expressions of φ n, m for n −1,2,…,5 and for all m ≥ 1 are given. A recurrence relation for φ n, m is also obtained. The values of φ n, m are tabulated for n = 1,…,5 and m = 0,1,…,10.

Chromatic Sums for Rooted Planar Triangulations, IV: The Case λ = ∞

SummaryThe chromial P(M, λ) of a planar near-triangulation M has the leading term λv(M), where v(M) is the number of vertices of M. The problem of finding the number of rooted planar near-triangulations of a given class S, all supposed to have the same number of vertices, can be regarded as a special case of the problem of finding chromatic sums. We can sum P(M, λ) over the members of S, divide by the appropriate power of λ and let λ → ∞. We thus get the sum of the coefficient of the leading term of P(M, λ) for all M ∈ S, that is we get the number of members of S.

Chromatic Sums for Rooted Planar Triangulations, III: The Case λ = 3

SummaryIn this paper we are chiefly concerned with the chromatic sums we have called l and h, with colour-number 3. In this case h can be interpreted as enumerating the rooted Eulerian triangulations with a given number of faces, and l as enumerating such triangulations with a given number of faces and a given valency for the root-vertex. The series h has been determined already, by summation from the formula enumerating even slicings [3]. However our formula for l does not seem to have been published before, though it could presumably be derived in a similar way.