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Discharging Method (discrete Mathematics)
The discharging method is a technique used to prove Lemma (mathematics), lemmas in structural graph theory. Discharging is most well known for its central role in the proof of the four color theorem. The discharging method is used to prove that every graph in a certain class contains some subgraph from a specified list. The presence of the desired subgraph is then often used to prove a graph coloring, coloring result. Most commonly, discharging is applied to planar graphs.. (Lecture text for Spring School on Combinatorics). Initially, a ''charge'' is assigned to each face and each vertex of the graph. The charges are assigned so that they sum to a small positive number. During the ''Discharging Phase'' the charge at each face or vertex may be redistributed to nearby faces and vertices, as required by a set of discharging rules. However, each discharging rule maintains the sum of the charges. The rules are designed so that after the discharging phase each face or vertex with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemma (mathematics)
In mathematics and other fields, a lemma (: lemmas or lemmata) is a generally minor, proven Theorem#Terminology, proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to mathematical proof, prove; however, a lemma can also turn out to be more important than originally thought. Etymology From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument. Comparison with theorem There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem#Terminology, Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof. Well-known lemmas Some powerful results in mathematics are known as lemmas, first named for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...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 of non-zero length (i.e., 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 doubts remain. The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken in a computer-aided proof. This came after many false proofs and mis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the Vertex (graph theory), vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an ''edge coloring'' assigns a color to each Edge (graph theory), edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each Face (graph theory), face (or region) so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graphs
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 a 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 calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Cases
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: # A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. # A proof of each of the cases. The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kenneth Appel
Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana–Champaign, solved the four-color theorem, one of the most famous problems in mathematics. They proved that any two-dimensional map, with certain limitations, can be filled in with four colors without any adjacent "countries" sharing the same color. The proof was controversial because it depended on thousands of computer calculations that could not be double-checked by hand, the first prominent example of such a process. Biography Appel was born in Brooklyn, New York, on October 8, 1932. He grew up in Queens, New York, and was the son of a Jewish couple, Irwin Appel and Lillian Sender Appel. He worked as an actuary for a brief time and then served in the U.S. Army for two years at Fort Benning, Georgia, and in Baumholder, Germany. In 1959, he finished his doctoral program at the University of Michigan, and he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Haken
Wolfgang Haken (; June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds. Biography Haken was born on June 21, 1928, in Berlin, Germany. His father was Werner Haken, a physicist who had Max Planck as a doctoral thesis advisor. In 1953, Haken earned a Ph.D. degree in mathematics from Christian-Albrechts-Universität zu Kiel (Kiel University) and married Anna-Irmgard von Bredow, who earned a Ph.D. degree in mathematics from the same university in 1959. In 1962, they left Germany so he could accept a position as visiting professor at the University of Illinois at Urbana-Champaign. He became a full professor in 1965, retiring in 1998. In 1976, together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved the four-color problem: they proved that any planar graph can be properly colored using at most four colors. Haken has introduced several ideas, including Haken ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neil Robertson (mathematician)
George Neil Robertson (born November 30, 1938) is a mathematician working mainly in topological graph theory, currently a distinguished professor emeritus at the Ohio State University. Education Robertson earned his B.Sc. from Brandon College in 1959 and his Ph.D. in 1969 at the University of Waterloo under his doctoral advisor William Tutte. Biography In 1969, Robertson joined the faculty of the Ohio State University, where he was promoted to Associate Professor in 1972 and Professor in 1984. He was a consultant with Bell Communications Research from 1984 to 1996. He has held visiting faculty positions in many institutions, most extensively at Princeton University from 1996 to 2001, and at Victoria University of Wellington, New Zealand, in 2002. He also holds an adjunct position at King Abdulaziz University in Saudi Arabia.. Research Robertson is known for his work in graph theory, and particularly for a long series of papers co-authored with Paul Seymour and published over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel P
Daniel commonly refers to: * Daniel (given name), a masculine given name and a surname * List of people named Daniel * List of people with surname Daniel * Daniel (biblical figure) * Book of Daniel, a biblical apocalypse, "an account of the activities and visions of Daniel" Daniel may also refer to: Arts and entertainment Literature * ''Daniel'' (Old English poem), an adaptation of the Book of Daniel * ''Daniel'', a 2006 novel by Richard Adams * ''Daniel'' (Mankell novel), 2007 Music * "Daniel" (Bat for Lashes song) (2009) * "Daniel" (Elton John song) (1973) * "Daniel", a song from '' Beautiful Creature'' by Juliana Hatfield * ''Daniel'' (album), a 2024 album by Real Estate Other arts and entertainment * ''Daniel'' (1983 film), by Sidney Lumet * ''Daniel'' (2019 film), a Danish film * Daniel (comics), a character in the ''Endless'' series Businesses * Daniel (department store), in the United Kingdom * H & R Daniel, a producer of English porcelain between 1827 and 18 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Seymour (mathematician)
Paul D. Seymour is a British mathematician known for his work in discrete mathematics, especially graph theory. He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website. Seymour is currently the Albert Baldwin Dod Professor of Mathematics at Princeton University. He won a Sloan Fellowship in 1983, and the Ostrowski Prize in 2003; and (sometimes with others) won the Fulkerson Prize in 1979, 1994, 2006 and 2009, and the Pólya Prize in 1983 and 2004. He received an honorary doctorate from the University of Waterloo in 2008, one from the Technical University of Denmark in 2013, and one from the École normale supérieure de Lyon in 2022. He was an invited speaker i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robin Thomas (mathematician)
Robin Thomas (August 22, 1962 – March 26, 2020) was a mathematician working in graph theory at the Georgia Institute of Technology. Thomas received his doctorate in 1985 from Charles University in Prague, Czechoslovakia (now the Czech Republic), under the supervision of Jaroslav Nešetřil. He joined the faculty at Georgia Tech in 1989, and became a Regents' Professor there, briefly serving as the department Chair. Personal life Thomas was married to Icelandic operations researcher Sigrún Andradóttir, also a professor at Georgia Tech. On March 26, 2020, he died of Amyotrophic Lateral Sclerosis at the age of 57 after 12 years of struggle with the illness. Awards Thomas was awarded the Fulkerson Prize for outstanding papers in discrete mathematics twice, in 1994 as co-author of a paper on the Hadwiger conjecture, and in 2009 for the proof of the strong perfect graph theorem. In 2011, he was awarded the Karel Janeček Foundation Neuron Prize for Lifetime Achievement in Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
