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Harborth's 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 (if true) 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. Special classes of graphs Although Harborth's conjecture is not known to be true for all planar graphs, it has been proven for several special kinds of planar graph. One class of graphs that have integral Fáry embeddings are the graphs that can be reduced to the empty graph by a sequence of operations of two types: *Removing a vertex of degree at most two. *Replacing a vertex of degree three by an edge between two of its neighbors. (If such an edge already exists, the degree three vertex can be remo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (graph Theory)
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. Degeneracy is also known as the ''k''-core number, width, and linkage, and is essentially the same as the coloring number or Szekeres–Wilf number (named after ). ''k''-degenerate graphs have also been called ''k''-inductive graphs. The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been (repeatedly) removed are called the ''k''-cores of the graph and the degeneracy of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Triangle
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ... length; any such rational triangle can be integrally rescaled (can have all sides multiplied by the same integer, namely a common multiple of their denominators) to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. However, other definitions of the term "rational triangle" also exist: In 1914 Carmichael used the term in the sense that we today use the term Heronian triangle; SomosSomos, M., "Rational triangles", http://grail.eecs.csuohio.edu/~somos/rattri.html uses it to refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Anning Theorem
The Erdős–Anning theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945. Rationality versus integrality Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers. For instance, the subset of points on a unit circle obtained by repeatedly rotating by the sharp angle in a 3–4–5 right triangle has this property. It forms a dense set in the circle. The (still unsolved) Erdős–Ulam problem asks whether there can exist a set of points at rational distances from each other that forms a dense set for the whole Euclidean plane. For any finite set ''S'' of points at rational distances from each other, it is possible to find a similar set of points at integer distances from each other, by expandi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete And Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Point Set
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 Fáry's theorem, straight-line drawing in which the vertices are all placed at points of ''S''. Bounds on the size of universal point sets When ''n'' is ten or less, there exist universal point sets with exactly ''n'' points, but for all ''n'' ≥ 15 additional points are required. Several authors have shown that subsets of the integer lattice of size ''O''(''n'') × ''O''(''n'') are universal. In particular, showed that a grid of (2''n'' − 3) × (''n'' − 1) points is universal, and reduced this to a triangular subset of an (''n'' − 1) × (''n'' − 1) grid, with ''n''2/2 − ''O''(''n'') points. By modifying the method of de Fraysseix et al., found an embedding of any planar graph into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X either ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Ulam Problem
In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam. Large point sets with rational distances The Erdős–Anning theorem states that a set of points with integer distances must either be finite or lie on a single line. However, there are other infinite sets of points with rational distances. For instance, on the unit circle, let ''S'' be the set of points :(\cos\theta,\sin\theta) where \theta is restricted to values that cause \tan\tfrac to be a rational number. For each such point, both \sin\tfrac and \cos\tfrac\theta 2 are themselves both rational, and if \theta and \varphi define two points in ''S'', then their distance is the rational number : \left, 2\sin\frac \theta 2 \cos\frac \varphi 2 -2\sin\frac \varphi 2 \cos\frac \theta 2 \. More generally, a circle with radius \rho contains a dense set of points at rational distances to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonian Network
In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction. Definition An Apollonian network may be formed, starting from a single triangle embedded in the Euclidean plane, by repeatedly selecting a triangular face of the embedding, adding a new vertex inside the face, and connecting the new vertex to each vertex of the face containing it. In this way, the triangle containing the new vertex is subdivided into three smaller triangles, which may in turn be subdivided in the same way. Examples The complete graphs on three and four vertices, and , are both Apollonian networks. is formed by starting with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the ''Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
