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Poussin Graph
In graph theory, the Poussin graph is a planar graph with 15 vertices and 39 edges. It is named after Charles Jean de la Vallée-Poussin. History In 1879, Alfred Kempe published a proof of the four color theorem, one of the big conjectures in graph theory. While the theorem is true, Kempe's proof is incorrect. Percy John Heawood illustrated it in 1890 with a counter-example, and de la Vallée-Poussin reached the same conclusion in 1896 with the Poussin graph. Kempe's (incorrect) proof is based on alternating chains, and as those chains prove useful in graph theory mathematicians remain interested in such counterexamples. More were found later: first, the Errera graph in 1921, then the Kittell graph in 1935, with 23 vertices, and finally two minimal counter-examples (the Soifer graph in 1997 and the Fritsch graph in 1998, both of order 9).Gethner, E. and Springer, W. M. II. « How False Is Kempe's Proof of the Four-Color Theorem? » Congr. Numer. 164, 159–175, 2003. Refere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Graph
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...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] |
Poussin Graph Tangled Kempe Chains
Nicolas Poussin (, , ; June 1594 – 19 November 1665) was the leading painter of the classical French Baroque style, although he spent most of his working life in Rome. Most of his works were on religious and mythological subjects painted for a small group of Italian and French collectors. He returned to Paris for a brief period to serve as First Painter to the King under Louis XIII and Cardinal Richelieu, but soon returned to Rome and resumed his more traditional themes. In his later years he gave growing prominence to the landscape in his paintings. His work is characterized by clarity, logic, and order, and favors line over color. Until the 20th century he remained a major inspiration for such classically-oriented artists as Jacques-Louis David, Jean-Auguste-Dominique Ingres and Paul Cézanne. Details of Poussin's artistic training are somewhat obscure. Around 1612 he traveled to Paris, where he studied under minor masters and completed his earliest surviving works. Hi ... [...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] |
Charles Jean De La Vallée-Poussin
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was '' Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as '' Carolus''. Some Germanic languages, for example Dutch and German, have retained the word in two separate senses. In the particular case of Dutch, ''Karel'' refers to the given name, whereas the noun ''kerel'' means "a bloke, fellow, man". Etymology The name's etymology is a Common Germanic noun ''*karilaz'' meaning "free man", which survives in English as churl (< Old English ''ċeorl''), which developed its depr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Kempe
Sir Alfred Bray Kempe FRS (6 July 1849 – 21 April 1922) was a mathematician best known for his work on linkages and the four colour theorem. Biography Kempe was the son of the Rector of St James's Church, Piccadilly, the Rev. John Edward Kempe. He was educated at St Paul's School, London and then studied at Trinity College, Cambridge, where Arthur Cayley was one of his teachers. He graduated BA (22nd wrangler) in 1872. Despite his interest in mathematics he became a barrister, specialising in the ecclesiastical law. He was knighted in 1913, the same year he became the Chancellor for the Diocese of London. He was also Chancellor of the dioceses of Newcastle, Southwell, St Albans, Peterborough, Chichester, and Chelmsford. He received the honorary degree DCL from the University of Durham and he was elected a Bencher of the Inner Temple in 1909. In 1876 he published his article ''On a General Method of describing Plane Curves of the nth degree by Linkwork,'' which presented a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four Color Theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet. It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubters remain. The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples (unlike the five color theorem, proved in the 1800s, which states that five colors are enough to color a map). To dispel any remaining doubts about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was publi ... [...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] |
Percy John Heawood
Percy John Heawood (8 September 1861 – 24 January 1955) was a British mathematician, who concentrated on graph colouring. Life He was the son of the Rev. John Richard Heawood of Newport, Shropshire, and his wife Emily Heath, daughter of the Rev. Joseph Heath of Wigmore, Herefordshire; and a first cousin of Oliver Lodge, whose mother Grace was also a daughter of Joseph Heath. He was educated at Queen Elizabeth's School, Ipswich, and matriculated at Exeter College, Oxford in 1880, graduating B.A. in 1883 and M.A. in 1887. Heawood spent his academic career at Durham University, where he was appointed Lecturer in 1885. He was, successively, Censor of St Cuthbert's Society between 1897 and 1901 succeeding Frank Byron Jevons in the role, Senior Proctor of the university from 1901, Professor in 1910 and Vice-Chancellor between 1926 and 1928. He was awarded an OBE, as Honorary Secretary of the Preservation Fund, for his part in raising £120,000 to prevent Durham Castle from col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kempe Chain
Kempe may refer to: * Kempe baronets, a title in the Baronetage of England * Kempe chain, part of the four-colour theorem * Kempe Fjord, King Christian X Land, Greenland * Kempe Glacier, Antarctica * Kempe Hill, former name of Camp Hill, West Midlands, England People with the surname * Alfred Kempe (1849–1922), English mathematician * Arnold E. Kempe (born 1927), American lawyer and politician * Carl Kempe (1884–1967), Swedish paper producer * Charles Eamer Kempe (1837–1907), English stained glass designer * C. Henry Kempe (1922–1984), American pediatrician who identified the Battered child syndrome * Kempe Gowda I (1513–69), Yelahanka chieftain, founded the city of Bangalore * Margery Kempe (c. 1373–after 1438), English autobiographer, religious pilgrim * Raymond J. Kempe (born 1931), American lawyer and politician * Rudolf Kempe (1910–76), German conductor * William Kempe (died c. 1603), English actor and morris dancer See also * Kemp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Errera Graph
In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof of the four color theorem; it was named after Errera by . Properties The Errera graph is planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4 and girth 3. All its vertices are of degree 5 or 6 and it is a 5- vertex-connected graph and a 5- edge-connected graph. The Errera graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 20, the group of symmetries of a decagon, including both rotations and reflections. The characteristic polynomial of the Errera graph is -(x^2-2 x-5) (x^2+x-1)^2 (x^3-4 x^2-9 x+10) (x^4+2 x^3-7 x^2-18 x-9)^2. Application to the four color theorem The four color theorem states that the vertices of every planar graph can be colored with four colors, so that no two adjacent vertices have equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kittell Graph
In the mathematical field of graph theory, the Kittell graph is a planar graph with 23 vertices and 63 edges. Its unique planar embedding has 42 triangular faces. The Kittell graph is named after Irving Kittell, who used it as a counterexample to Alfred Kempe's flawed proof of the four-color theorem. Simpler counterexamples include the Errera graph In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof of the four color theorem; it was named after Errera by . Pro ... and Poussin graph (both published earlier than Kittell) and the Fritsch graph and Soifer graph. References Individual graphs Planar graphs {{graph-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
