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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 chromatic ... [...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] |
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] |
Chi Bounded
__NOTOC__ Chi may refer to: __NOTOC__ Greek *Chi (letter) (Χ or χ), the twenty-second letter of the Greek alphabet Chinese * ''Chi'' (length) (å°º), a traditional unit of length, about â…“ meter *Chi (mythology) (èž), a dragon *Chi (surname) (æ± , pinyin: ''chÃ'') * ''Ch'i'' or ''qi'' (æ°£), "energy force" *Chinese language (ISO 639-2 code "chi") *Ji (surname), various surnames written Chi in Wade–Giles Arts and entertainment * ''Chi'' (2013 film), a Canadian documentary film * ''Chi'' (2019 film), a Burmese drama film *'' Chi: On the Movements of the Earth'', a manga series by Uoto *''The Chi'', an American drama television series * Chi (''Chobits''), a character in the manga series ''Chobits'' *Sailor Chi, a character in the manga series ''Sailor Moon'' *Chi, a character in the manga series ''Chi's Sweet Home'' *"Chi", a song by Korn from the 1996 album ''Life Is Peachy'' Science and mathematics *Chi, the hyperbolic cosine integral *CHI (conference) for SIGCHI and its an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boxicity
In graph theory, boxicity is a graph invariant, introduced by Fred S. Roberts in 1969. The boxicity of a graph is the minimum dimension in which a given graph can be represented as an intersection graph of axis-parallel boxes. That is, there must exist a one-to-one correspondence between the vertices of the graph and a set of boxes, such that two boxes intersect if and only if there is an edge connecting the corresponding vertices. Examples The figure shows a graph with six vertices, and a representation of this graph as an intersection graph of rectangles (two-dimensional boxes). This graph cannot be represented as an intersection graph of boxes in any lower dimension, so its boxicity is two. showed that the graph with 2''n'' vertices formed by removing a perfect matching from a complete graph on 2''n'' vertices has boxicity exactly ''n'': each pair of disconnected vertices must be represented by boxes that are separated in a different dimension than each other pair. A box re ... [...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] |
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] |
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] |
Even-hole-free Graph
In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices. More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs. demonstrated that every even-cycle-free graph contains a bisimplicial vertex (a vertex whose neighborhood is the union of two cliques), which settled a conjecture by Reed. The proof was later shown to be flawed by , who gave a correct proof. Recognition gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in (n^) time. later improved this to (n^). and improved this to (n^) time. The best currently known algorithm is given by which runs in (n^9) time. While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex. It is unknown whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrangement Of Lines
In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons, the ''cells'' of the arrangement, line segments and rays, the ''edges'' of the arrangement, and points where two or more lines cross, the ''vertices'' of the arrangement. When considered in the projective plane rather than in the Euclidean plane, every two lines cross, and an arrangement is the projective dual to a finite set of points. Arrangements of lines have also been considered in the hyperbolic plane, and generalized to ''pseudolines'', curves that have similar topological properties to lines. The initial study of arrangements has been attributed to an 1826 paper by Jakob Steiner. An arrangement is said to be ''simple'' when at most two lines cross at each vertex, and ''simplicial'' when all cells are triangles (including the unbounded cells, as subsets of the projective plane). There are three kno ... [...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] |
