Thrackle

Last updated

A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. In the latter case, the crossing must be transverse . [1]

Contents

Linear thrackles

A linear thrackle is a thrackle drawn in such a way that its edges are straight line segments. Every linear thrackle has at most as many edges as vertices, a fact that was observed by Paul Erdős. Erdős observed that, if a vertex v is connected to three or more edges vw, vx, and vy in a linear thrackle, then at least one of those edges lies on a line that separates two other edges; without loss of generality assume that vw is such an edge, with x and y lying in opposite closed halfspaces bounded by line vw. Then, w must have degree one, because no other edge than vw can touch both vx and vy. Removing w from the thrackle produces a smaller thrackle, without changing the difference between the numbers of edges and vertices. On the other hand, if every vertex has at most two neighbors, then by the handshaking lemma the number of edges is at most the number of vertices. [2] Based on Erdős' proof, one can infer that every linear thrackle is a pseudoforest. Every cycle of odd length may be arranged to form a linear thrackle, but this is not possible for an even-length cycle, because if one edge of the cycle is chosen arbitrarily then the other cycle vertices must lie alternatingly on opposite sides of the line through this edge.

Micha Perles provided another simple proof that linear thrackles have at most n edges, based on the fact that in a linear thrackle every edge has an endpoint at which the edges span an angle of at most 180°, and for which it is the most clockwise edge within this span. For, if not, there would be two edges, incident to opposite endpoints of the edge and lying on opposite sides of the line through the edge, which could not cross each other. But each vertex can only have this property with respect to a single edge, so the number of edges is at most equal to the number of vertices. [3]

As Erdős also observed, the set of pairs of points realizing the diameter of a point set must form a linear thrackle: no two diameters can be disjoint from each other, because if they were then their four endpoints would have a pair at farther distance apart than the two disjoint edges. For this reason, every set of n points in the plane can have at most n diametral pairs, answering a question posed in 1934 by Heinz Hopf and Erika Pannwitz. [4] Andrew Vázsonyi conjectured bounds on the number of diameter pairs in higher dimensions, generalizing this problem. [2]

In computational geometry, the method of rotating calipers can be used to form a linear thrackle from any set of points in convex position, by connecting pairs of points that support parallel lines tangent to the convex hull of the points. [5] This graph contains as a subgraph the thrackle of diameter pairs. [6]

An enumeration of linear thrackles may be used to solve the biggest little polygon problem, of finding an n-gon with maximum area relative to its diameter. [7]

Thrackle conjecture

Question, Web Fundamentals.svgUnsolved problem in mathematics:
Can a thrackle have more edges than vertices?
(more unsolved problems in mathematics)
A thrackle embedding of a 6-cycle graph. 6-cycle thrackle.png
A thrackle embedding of a 6-cycle graph.

John H. Conway conjectured that, in any thrackle, the number of edges is at most equal to the number of vertices. Conway himself used the terminology paths and spots (for edges and vertices respectively), so Conway's thrackle conjecture was originally stated in the form every thrackle has at least as many spots as paths. Conway offered a $1000 prize for proving or disproving this conjecture, as part of a set of prize problems also including Conway's 99-graph problem, the minimum spacing of Danzer sets, and the winner of Sylver coinage after the move 16. [8]

Equivalently, the thrackle conjecture may be stated as every thrackle is a pseudoforest. More specifically, if the thrackle conjecture is true, the thrackles may be exactly characterized by a result of Woodall: they are the pseudoforests in which there is no cycle of length four and at most one odd cycle. [1] [9]

It has been proven that every cycle graph other than C4 has a thrackle embedding, which shows that the conjecture is sharp. That is, there are thrackles having the same number of spots as paths. At the other extreme, the worst-case scenario is that the number of spots is twice the number of paths; this is also attainable.

The thrackle conjecture is known to be true for thrackles drawn in such a way that every edge is an x-monotone curve, crossed at most once by every vertical line. [3]

Known bounds

Lovász, Pach & Szegedy (1997) proved that every bipartite thrackle is a planar graph, although not drawn in a planar way. [1] As a consequence, they show that every thrackleable graph with n vertices has at most 2n  3 edges. Since then, this bound has been improved several times. First, it was improved to 3(n  1)/2, [10] and another improvement led to a bound of roughly 1.428n. [11] Moreover, the method used to prove the latter result yields for any ε > 0 a finite algorithm that either improves the bound to (1 + ε)n or disproves the conjecture. The current record is due to Fulek & Pach (2017), who proved a bound of 1.3984n. [12]

If the conjecture is false, a minimal counterexample would have the form of two even cycles sharing a vertex. [9] Therefore, to prove the conjecture, it would suffice to prove that graphs of this type cannot be drawn as thrackles.

Related Research Articles

This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.

Edge coloring an assignment of colors to the edges of a graph so that no two edges that share an endpoint have the same color as each other

In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

Erdős–Faber–Lovász conjecture conjecture about coloring graphs formed by combining complete graphs

In graph theory, the Erdős–Faber–Lovász conjecture is an unsolved problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972. It says:

In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding.

Sylvester–Gallai theorem theorem that every finite set of points in the plane, not all collinear, has a line through exactly two points

The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either

Degree (graph theory) number of edges incident to a given vertex in a node-link graph

In graph theory, the degree of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex is denoted or . The maximum degree of a graph , denoted by , and the minimum degree of a graph, denoted by , are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.

In mathematics, the Erdős–Burr conjecture was a problem concerning the Ramsey number of sparse graphs. The conjecture is named after Paul Erdős and Stefan Burr, and is one of many conjectures named after Erdős; it states that the Ramsey number of graphs in any sparse family of graphs should grow linearly in the number of vertices of the graph.

Geometric graph theory type of graph theory

Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are allowed to be arbitrary continuous curves connecting the vertices, thus it is "the theory of geometric and topological graphs".

Planarity is a puzzle computer game by John Tantalo, based on a concept by Mary Radcliffe at Western Michigan University. The name comes from the concept of planar graphs in graph theory; these are graphs that can be embedded in the Euclidean plane so that no edges intersect. By Fáry's theorem, if a graph is planar, it can be drawn without crossings so that all of its edges are straight line segments. In the planarity game, the player is presented with a circular layout of a planar graph, with all the vertices placed on a single circle and with many crossings. The goal for the player is to eliminate all of the crossings and construct a straight-line embedding of the graph by moving the vertices one by one into better positions.

Pseudoforest undirected graph in which every connected component has at most one cycle

In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest.

Crossing number (graph theory) the smallest number of edge crossings possible in a drawing of a node-link graph

In graph theory, the crossing numbercr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing.

In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of O(√n) vertices from an n-vertex graph can partition the graph into disjoint subgraphs each of which has at most 2n/3 vertices.

Halin graph a type of planar graph formed from a tree by adding a cycle of edges through its leaves

In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the plane so none of its edges cross, and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.

In the study of graph algorithms, an implicit graph representation is a graph whose vertices or edges are not represented as explicit objects in a computer's memory, but rather are determined algorithmically from some more concise input.

Degeneracy (graph theory) mathematical concept

In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.

Topological graph

In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other (without crossing). A topological graph is also called a drawing of a graph.

In graph drawing, a universal point set of order n is a set S of points in the Euclidean plane with the property that every n-vertex planar graph has a straight-line drawing in which the vertices are all placed at points of S.

Harborths conjecture

In mathematics, Harborth's conjecture states that every planar graph has a planar drawing in which every edge is a straight segment of integer length. This conjecture is named after Heiko Harborth, and would strengthen Fáry's theorem on the existence of straight-line drawings for every planar graph. For this reason, a drawing with integer edge lengths is also known as an integral Fáry embedding. Despite much subsequent research, Harborth's conjecture remains unsolved.

In the mathematics of graph drawing, the crossing number inequality or crossing lemma gives a lower bound on the minimum number of crossings of a given graph, as a function of the number of edges and vertices of the graph. It states that, for graphs where the number e of edges is sufficiently larger than the number n of vertices, the crossing number is at least proportional to e3/n2.

Penny graph contact graph of unit circles

In geometric graph theory, a penny graph is a contact graph of unit circles. That is, it is an undirected graph whose vertices can be represented by unit circles, with no two of these circles crossing each other, and with two adjacent vertices if and only if they are represented by tangent circles. More simply, they are the graphs formed by arranging pennies in a non-overlapping way on a flat surface, making a vertex for each penny, and making an edge for each two pennies that touch.

References

  1. 1 2 3 Lovász, L.; Pach, J.; Szegedy, M. (1997), "On Conway's thrackle conjecture", Discrete and Computational Geometry , 18 (4): 369–376, doi: 10.1007/PL00009322 , MR   1476318 . A preliminary version of these results was reviewed in O'Rourke, J. (1995), "Computational geometry column 26", ACM SIGACT News, 26 (2): 15–17, arXiv: cs/9908007 , doi:10.1145/202840.202842 .
  2. 1 2 Erdős, P. (1946), "On sets of distances of n points" (PDF), American Mathematical Monthly , 53: 248–250, doi:10.2307/2305092 .
  3. 1 2 Pach, János; Sterling, Ethan (2011), "Conway's conjecture for monotone thrackles", American Mathematical Monthly , 118 (6): 544–548, doi:10.4169/amer.math.monthly.118.06.544, MR   2812285 .
  4. Hopf, H.; Pannwitz, E. (1934), "Aufgabe Nr. 167", Jahresbericht der Deutschen Mathematiker-Vereinigung, 43: 114.
  5. Eppstein, David (May 1995), "The Rotating Caliper Graph", The Geometry Junkyard
  6. For the fact that the rotating caliper graph contains all diameter pairs, see Shamos, Michael (1978), Computational Geometry (PDF), Doctoral dissertation, Yale University. For the fact that the diameter pairs form a thrackle, see, e.g., Pach & Sterling (2011).
  7. Graham, R. L. (1975), "The largest small hexagon" (PDF), Journal of Combinatorial Theory , Series A, 18: 165–170, doi:10.1016/0097-3165(75)90004-7 .
  8. Conway, John H., Five $1,000 Problems (Update 2017) (PDF), Online Encyclopedia of Integer Sequences, retrieved 2019-02-12
  9. 1 2 Woodall, D. R. (1969), "Thrackles and deadlock", in Welsh, D. J. A. (ed.), Combinatorial Mathematics and Its Applications, Academic Press, pp. 335–348, MR   0277421 .
  10. Cairns, G.; Nikolayevsky, Y. (2000), "Bounds for generalized thrackles", Discrete and Computational Geometry, 23 (2): 191–206, doi: 10.1007/PL00009495 , MR   1739605 .
  11. Fulek, R.; Pach, J. (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, arXiv: 1002.3904 , doi:10.1007/978-3-642-18469-7_21, MR   2785903 .
  12. Fulek, R.; Pach, J. (2017), Thrackles: An Improved Upper Bound, International Symposium on Graph Drawing and Network Visualization, pp. 160–166, arXiv: 1708.08037 , doi:10.1007/978-3-319-73915-1_14 .