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Hadwiger Conjecture (graph Theory)
In graph theory, the Hadwiger conjecture states that if G is loopless and has no K_t minor then its chromatic number satisfies It is known to be true for The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field. In more detail, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single vertex produces a complete graph K_k on k vertices as a minor This conjecture, a far-reaching generalization of the four-color problem, was made by Hugo Hadwiger in 1943 and is still unsolved. call it "one of the deepest unsolved problems in graph theory." Equivalent forms An equivalent form of the Hadwiger conjecture (the contrapositive of the form stated above ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the 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 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 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 a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contrapositive
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped. Conditional statement P \rightarrow Q. In formulas: the contrapositive of P \rightarrow Q is \neg Q \rightarrow \neg P . If ''P'', Then ''Q''. — If not ''Q'', Then not ''P''. ''"''If ''it is raining,'' then ''I wear my coat" —'' "If ''I don't wear my coat,'' then ''it isn't raining."'' The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true. The contrapositive ( \neg Q \rightarrow \neg P ) can be compared with three other statements: ;Inversion (the inverse), \neg P \rightarrow \neg Q:"If ''it is not raining,'' then ''I don't wear my coat''." Unlike the contrapositive, the inverse's truth value is not at all depe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Ladder
In graph theory, the Möbius ladder , for even numbers , is formed from an by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of (the utility graph ), has exactly four-cycles which link together by their shared edges to form a topological Möbius strip. Möbius ladders were named and first studied by . Properties For every even , the Möbius ladder is a nonplanar apex graph, meaning that it cannot be drawn without crossings in the plane but removing one vertex allows the remaining graph to be drawn without crossings. These graphs have crossing number one, and can be embedded without crossings on a torus or projective plane. Thus, they are examples of toroidal graphs. explores embeddings of these graphs onto higher genus surfaces. Möbius ladders are vertex-transitive – they have symmetries taking any vertex to any other vertex – but (with the exceptions of and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique-sum
In graph theory, a branch of mathematics, a clique-sum is a way of combining two graphs by gluing them together at a clique, analogous to the connected sum operation in topology. If two graphs ''G'' and ''H'' each contain cliques of equal size, the clique-sum of ''G'' and ''H'' is formed from their disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, and then possibly deleting some of the clique edges. A ''k''-clique-sum is a clique-sum in which both cliques have at most ''k'' vertices. One may also form clique-sums and ''k''-clique-sums of more than two graphs, by repeated application of the two-graph clique-sum operation. Different sources disagree on which edges should be removed as part of a clique-sum operation. In some contexts, such as the decomposition of chordal graphs or strangulated graphs, no edges should be removed. In other contexts, such as the SPQR-tree decomposition of graphs into their 3-vertex-connected components, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klaus Wagner (mathematician)
Klaus Wagner (March 31, 1910 – February 6, 2000) was a German mathematician known for his contributions to graph theory. Education and career Wagner studied topology at the University of Cologne under the supervision of who had been a student of Issai Schur. Wagner received his Ph.D. in 1937, with a dissertation concerning the Jordan curve theorem and four color theorem, and taught at Cologne for many years himself. In 1970, he moved to the University of Duisburg, where he remained until his retirement in 1978. Graph minors Wagner is known for his contributions to graph theory and particularly the theory of graph minors, graphs that can be formed from a larger graph by contracting and removing edges. Wagner's theorem characterizes the planar graphs as exactly those graphs that do not have as a minor either a complete graph ''K''5 on five vertices or a complete bipartite graph ''K''3,3 with three vertices on each side of its bipartition. That is, these two graphs are the onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wagner's Theorem
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither ''K''5 (the complete graph on five vertices) nor ''K''3,3 (the utility graph, a complete bipartite graph on six vertices). This was one of the earliest results in the theory of graph minors and can be seen as a forerunner of the Robertson–Seymour theorem. Definitions and statement A planar embedding of a given graph is a drawing of the graph in the Euclidean plane, with points for its vertices and curves for its edges, in such a way that the only intersections between pairs of edges are at a common endpoint of the two edges. A minor of a given graph is another graph formed by deleting vertices, deleting edges, and contracting edges. When an edge is contracted, its two endpoints are merged to form a single vertex. In some versions of graph minor theory the graph r ... [...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] |
Series–parallel Graph
In graph theory, series–parallel graphs are graphs with two distinguished vertices called ''terminals'', formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits. Definition and terminology In this context, the term graph means multigraph. There are several ways to define series–parallel graphs. The following definition basically follows the one used by David Eppstein. A two-terminal graph (TTG) is a graph with two distinguished vertices, ''s'' and ''t'' called ''source'' and ''sink'', respectively. The parallel composition ''Pc = Pc(X,Y)'' of two TTGs ''X'' and ''Y'' is a TTG created from the disjoint union of graphs ''X'' and ''Y'' by merging the sources of ''X'' and ''Y'' to create the source of ''Pc'' and merging the sinks of ''X'' and ''Y'' to create the sink of ''Pc''. The series composition ''Sc = Sc(X,Y)'' of two TTGs ''X'' and ''Y'' is a TTG created from the disjoint union of graphs ''X'' and ''Y'' ... [...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] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadwiger Number
In graph theory, the Hadwiger number of an undirected graph is the size of the largest complete graph that can be obtained by contracting edges of . Equivalently, the Hadwiger number is the largest number for which the complete graph is a minor of , a smaller graph obtained from by edge contractions and vertex and edge deletions. The Hadwiger number is also known as the contraction clique number of or the homomorphism degree of . It is named after Hugo Hadwiger, who introduced it in 1943 in conjunction with the Hadwiger conjecture, which states that the Hadwiger number is always at least as large as the chromatic number of . The graphs that have Hadwiger number at most four have been characterized by . The graphs with any finite bound on the Hadwiger number are sparse, and have small chromatic number. Determining the Hadwiger number of a graph is NP-hard but fixed-parameter tractable. Graphs with small Hadwiger number A graph has Hadwiger number at most two if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forbidden Graph Characterization
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph and the complete bipartite graph . For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which case it does not belong to the planar graphs). Definition More generally, a forbidden grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
