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Erdős–Gyárfás Conjecture
In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by the prolific mathematician Paul Erdős and his collaborator András Gyárfás, states that every graph with minimum degree 3 contains a simple cycle whose length is a power of two. Erdős offered a prize of $100 for proving the conjecture, or $50 for a counterexample; it is one of many conjectures of Erdős. If the conjecture is false, a counterexample would take the form of a graph with minimum degree three having no power-of-two cycles. It is known through computer searches of Gordon Royle and Klas Markström that any counterexample must have at least 17 vertices, and any cubic counterexample must have at least 30 vertices. Markström's searches found four graphs on 24 vertices in which the only power-of-two cycles have 16 vertices. One of these four graphs is planar; however, the Erdős–Gyárfás conjecture is now known to be true for the special case of 3-connected cubic planar graphs Weaker resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gordon Royle
Gordon F. Royle is a professor at the School of Mathematics and Statistics at The University of Western Australia. Royle is the co-author (with Chris Godsil) of the book ''Algebraic Graph Theory'' (Springer Verlag, 2001, ). Royle is also known for his research into the mathematics of Sudoku and his search for the Sudoku puzzle with the smallest number of entries that has a unique solution. Royle earned his Ph.D. in 1987 from the University of Western Australia under the supervision of Cheryl Praeger Cheryl Elisabeth Praeger (born 7 September 1948, Toowoomba, Queensland) is an Australian mathematician. Praeger received BSc (1969) and MSc degrees from the University of Queensland (1974), and a doctorate from the University of Oxford in 197 ... and Brendan McKay. References {{DEFAULTSORT:Royle, Gordon Living people Australian mathematicians Graph theorists University of Western Australia alumni University of Western Australia faculty Year of birth missing (livi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star (graph Theory)
In graph theory, a star is the complete bipartite graph a tree with one internal node and leaves (but no internal nodes and leaves when ). Alternatively, some authors define to be the tree of order with maximum diameter 2; in which case a star of has leaves. A star with 3 edges is called a claw. The star is edge-graceful when is even and not when is odd. It is an edge-transitive matchstick graph, and has diameter 2 (when ), girth ∞ (it has no cycles), chromatic index , and chromatic number 2 (when ). Additionally, the star has large automorphism group, namely, the symmetric group on letters. Stars may also be described as the only connected graphs in which at most one vertex has degree greater than one. Relation to other graph families Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claw-free Graph
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph ''K''1,3 (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, and the characterization of claw-free perfect graphs., p. 88. They are the subject of hundreds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Logarithm
In computer science, the iterated logarithm of n, written n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition is the result of this recurrence relation: : \log^* n := \begin 0 & \mbox n \le 1; \\ 1 + \log^*(\log n) & \mbox n > 1 \end On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent: :\log^* n = \lceil \mathrm _e(n) \rceil i.e. the base ''b'' iterated logarithm is \log^* n = y if n lies within the interval ^b on the ''x''-axis. In computer science, is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base 2) instead of the natural logarithm (with base ''e''). Mathematically, the iterated logarithm is well-defined for any base greater than e^ \approx 1.444667, not only for base 2 and base ''e''. Analysis of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...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] |
List Of Conjectures By Paul Erdős
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects, and in many cases Erdős offered monetary rewards for solving them. Unsolved * The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3. * The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set. * The Erdős–Mollin–Walsh conjecture on consecutive triples of powerful numbers. * The Erdős–Selfridge conjecture that a covering system with distinct moduli contains at least one even modulus. * The Erdős–Straus conjecture on the Diophantine equation 4/''n'' = 1/''x'' + 1/''y'' + 1/''z''. * The Erdős conjecture on arithmetic progressions in sequences with divergent sums of reciprocals. * The Erdős–Szekeres conjecture on the number of points needed to ensure that a point se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of Two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system. Computer science Two to the exponent of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
