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χ-bounded
In graph theory, a \chi-bounded family \mathcal of graphs is one for which there is some function f such that, for every integer t the graphs in \mathcal with t=\omega(G) ( clique number) can be colored with at most f(t) colors. The function f(t) is called a \chi-binding function for \mathcal. These concepts and their notations were formulated by András Gyárfás. The use of the Greek letter chi in the term \chi-bounded is based on the fact that the chromatic number of a graph G is commonly denoted \chi(G). An overview of the area can be found in a survey of Alex Scott and Paul Seymour. Nontriviality It is not true that the family of all graphs is \chi-bounded. As , and showed, there exist triangle-free graphs of arbitrarily large chromatic number, so for these graphs it is not possible to define a finite value of f(2). Thus, \chi-boundedness is a nontrivial concept, true for some graph families and false for others. Specific classes Every class of graphs of bounded chro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisimplicial Vertex
In graph theory, a simplicial vertex v is a vertex (graph theory), vertex whose neighborhood (graph theory), closed neighborhood N_[v] in a graph G forms a clique (graph theory), clique, where every pair of neighbors is adjacent to each other. A vertex of a graph is bisimplicial if the set of it and its neighbours is the union of two cliques, and is -simplicial if the set is the union of cliques. A vertex is co-simplicial if its non-neighbours form an independent set (graph theory), independent set. Addario-Berry et al. demonstrated that every even-hole-free graph (or more specifically, even-cycle-free graph, as 4-cycles are also excluded here) contains a bisimplicial vertex, which settled a conjecture by Reed. The proof was later shown to be flawed by Chudnovsky & Seymour, who gave a correct proof. Due to this property, the family of all even-cycle-free graphs is χ-bounded, \chi-bounded. See also *Even-hole-free graph *χ-bounded, \chi-bounded family of graphs References ... [...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] |
Gyárfás–Sumner Conjecture
In graph theory, the Gyárfás–Sumner conjecture asks whether, for every tree T and complete graph K, the graphs with neither T nor K as induced subgraphs can be properly colored using only a constant number of colors. Equivalently, it asks whether the T-free graphs are \chi-bounded. It is named after András Gyárfás and David Sumner, who formulated it independently in 1975 and 1981 respectively. It remains unproven. In this conjecture, it is not possible to replace T by a graph with cycles. As Paul Erdős and András Hajnal have shown, there exist graphs with arbitrarily large chromatic number and, at the same time, arbitrarily large girth. Using these graphs, one can obtain graphs that avoid any fixed choice of a cyclic graph and clique (of more than two vertices) as induced subgraphs, and exceed any fixed bound on the chromatic number. The conjecture is known to be true for certain special choices of T, including paths, stars A star is a luminous spheroid of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique-width
In graph theory, the clique-width of a graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be small for dense graphs. It is defined as the minimum number of labels needed to construct by means of the following 4 operations : #Creation of a new vertex with label (denoted by ) #Disjoint union of two labeled graphs and (denoted by G \oplus H) #Joining by an edge every vertex labeled to every vertex labeled (denoted by ), where #Renaming label to label (denoted by ) Graphs of bounded clique-width include the cographs and distance-hereditary graphs. Although it is NP-hard to compute the clique-width when it is unbounded, and unknown whether it can be computed in polynomial time when it is bounded, efficient approximation algorithms for clique-width are known. Based on these algorithms and on Courcelle's theorem, many graph optimization problems that are NP-hard for arbitrary graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Graph
In graph theory, a circle graph is the intersection graph of a Chord diagram (mathematics), chord diagram. That is, it is an undirected graph whose vertices can be associated with a finite system of Chord (geometry), chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other. Algorithmic complexity After earlier polynomial time algorithms, presented an algorithm for recognizing circle graphs in near-linear time. Their method is slower than linear by a factor of the inverse Ackermann function, and is based on lexicographic breadth-first search. The running time comes from a method for maintaining the split decomposition of a graph incrementally, as vertices are added, used as a subroutine in the algorithm. A number of other problems that are NP-complete on general graphs have polynomial time algorithms when restricted to circle graphs. For instance, showed that the treewidth of a circle graph can be determined, and an optim ... [...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 (graph theory), claw as an induced subgraph. A claw is another name for the complete bipartite graph K_ (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 (graph theory), neighborhood of any vertex (graph theory), vertex is the complement (graph theory), 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, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a 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 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 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 (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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
András Gyárfás
András Gyárfás (born 1945) is a Hungarian mathematician who specializes in the study of graph theory. He is famous for two conjectures: * Together with Paul Erdős he conjectured what is now called the Erdős–Gyárfás conjecture which states that any graph with minimum degree 3 contains a cycle whose length is a power of two. * He and David Sumner independently formulated the Gyárfás–Sumner conjecture according to which, for every tree ''T'', the ''T''-free graphs are χ-bounded In graph theory, a \chi-bounded family \mathcal of graphs is one for which there is some function f such that, for every integer t the graphs in \mathcal with t=\omega(G) ( clique number) can be colored with at most f(t) colors. The function f(t) .... Gyárfás began working as a researcher for the Computer and Automation Research Institute of the Hungarian Academy of Sciences in 1968. He earned a candidate degree in 1980, and a doctorate (Dr. Math. Sci.) in 1992. He won the Géza Grü ... [...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] |
Perfect Graph
In graph theory, a perfect graph is a Graph (discrete mathematics), graph in which the Graph coloring, chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices. The perfect graphs include many important families of graphs and serve to unify results relating Graph coloring, colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time, despite their greater complexity for non-perfect graphs. In addition, several important minimax theorems in combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, KÅ‘nig's theorem (gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Graph
In graph theory, a string graph is an intersection graph of curves in the plane; each curve is called a "string". Given a graph , is a string graph if and only if there exists a set of curves, or strings, such that the graph having a vertex for each curve and an edge for each intersecting pair of curves is isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ... to . Background described a concept similar to string graphs as they applied to genetic structures. In that context, he also posed the specific case of intersecting intervals on a line, namely the now-classical family of interval graphs. Later, specified the same idea to electrical networks and printed circuits. The mathematical study of string graphs began with the paper and through a collaboration between Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
