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Triangle-free
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with edges is triangle-free in time . Another approach is to find the trace of , where is the adjacency matrix of the graph. The trace is zero if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grötzsch's Theorem
In the mathematical field of graph theory, Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors. According to the four-color theorem, every graph that can be drawn in the plane without edge crossings can have its vertices colored using at most four different colors, so that the two endpoints of every edge have different colors, but according to Grötzsch's theorem only three colors are needed for planar graphs that do not contain three mutually adjacent vertices. History The theorem is named after German mathematician Herbert Grötzsch, who published its proof in 1959. Grötzsch's original proof was complex. attempted to simplify it but his proof was erroneous.. In 2003, Carsten Thomassen derived an alternative proof from another related theorem: every planar graph with girth at least five is 3-list-colorable. However, Grötzsch's theorem itself does not extend from coloring to list coloring: there exist triangle-free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mycielskian
In the mathematical area of graph theory, the Mycielskian or Mycielski graph of an undirected graph is a larger graph formed from it by a construction of . The construction preserves the property of being triangle-free but increases the chromatic number; by applying the construction repeatedly to a triangle-free starting graph, Mycielski showed that there exist triangle-free graphs with arbitrarily large chromatic number. Construction Let the ''n'' vertices of the given graph ''G'' be ''v''1, ''v''2, . . . , ''v''n. The Mycielski graph μ(''G'') contains ''G'' itself as a subgraph, together with ''n''+1 additional vertices: a vertex ''u''''i'' corresponding to each vertex ''v''''i'' of ''G'', and an extra vertex ''w''. Each vertex ''u''''i'' is connected by an edge to ''w'', so that these vertices form a subgraph in the form of a star ''K''1,''n''. In addition, for each edge ''v''''i''''v''''j'' of ''G'', the Mycielski graph includes two edges, ''u''''i''''v''''j'' and ''v''''i' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Median Graph
In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices ''a'', ''b'', and ''c'' have a unique ''median'': a vertex ''m''(''a'',''b'',''c'') that belongs to shortest paths between each pair of ''a'', ''b'', and ''c''. The concept of median graphs has long been studied, for instance by or (more explicitly) by , but the first paper to call them "median graphs" appears to be . As Chung, Graham, and Saks write, "median graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature".. In phylogenetics, the Buneman graph representing all maximum parsimony Phylogenetic tree, evolutionary trees is a median graph. Median graphs also arise in social choice theory: if a set of alternatives has the structure of a median graph, it is possible to derive in an unambiguous way a majority preference among them. Additional surveys of median graphs are given by , , and . Examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
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] |
Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a -cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Image:Petersen1 tiny.svg, The Petersen graph has a girth of 5 Image:Heawood_Graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighbourhood (graph Theory)
In graph theory, an adjacent vertex of a vertex in a graph is a vertex that is connected to by an edge. The neighbourhood of a vertex in a graph is the subgraph of induced by all vertices adjacent to , i.e., the graph composed of the vertices adjacent to and all edges connecting vertices adjacent to . The neighbourhood is often denoted or (when the graph is unambiguous) . The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include itself, and is more specifically the open neighbourhood of ; it is also possible to define a neighbourhood in which itself is included, called the closed neighbourhood and denoted by . When stated without any qualification, a neighbourhood is assumed to be open. Neighbourhoods may be used to represent graphs in computer algorithms, via the adjacency list and adjacency matrix representations. Neighbourhoods are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán's Theorem
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph. An example of an n-vertex graph that does not contain any (r+1)-vertex clique K_ may be formed by partitioning the set of n vertices into r parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph T(n,r). Turán's theorem states that the Turán graph has the largest number of edges among all -free -vertex graphs. Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey Number
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges of ... [...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] |
Maximal Independent Set
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property. For example, in the graph , a path with three vertices , , and , and two edges and , the sets and are both maximally independent. The set is independent, but is not maximal independent, because it is a subset of the larger independent set In this same graph, the maximal cliques are the sets and A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets. A graph may have many MISs of widely varying sizes; the largest, or possibly several equally large, MISs of a graph is called a maximum independent set. The graphs in which all maximal independent sets have the same siz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biclique K 5 5
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
