Graph coloring - Graph coloring A 3-coloring of a graph Many terms used in this article are defined in the Glossary of graph theory. A coloring of a Graph is an assignment of colors to the vertices such that no two adjacent vertices are assigned the same color. Here, "adjacent" means sharing the same edge. Graph coloring with colors is equivalent to the problem of partioning the vertex set into independent sets. The problem of coloring a graph has found a number of applications such as scheduling, register allocation in a microprocessor, frequency assignment in mobile radios, and pattern matching. In general, techniques for graph coloring concentrate on finding the least number of colors needed to color the graph ie. its chromatic number . For example the chromatic number.
Graph theory - Graph theory Graph theory is the branch of mathematics that examines the properties of graphs. A graph with 6 vertices and 7 edges. Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs). Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges). For more and formal definitions, see Glossary of graph theory and Graph (mathematics). Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, i.e. numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) we have a.
List of graph theory topics - List of graph theory topics This is a list of graph theory topics, by Wikipedia page. See glossary of graph theory for basic terminology Table of contents showTocToggle("show","hide") 1 Examples 2 Coloring 3 Paths and cycles 4 Trees 5 Mazes 6 Algorithms 7 Other topics 8 Hypergraphs Examples Complete graph Petersen graph Coloring Graph coloring Bipartite graphs Disperser Expander Extractor Four color theorem Tait's conjecture Ramsey's theorem Paths and cycles Seven bridges of Königsberg Shortest path problem Dijkstra's algorithm Open shortest path first Flooding algorithm Route inspection problem Hamiltonian cycle problem, Hamiltonian path Knight's Tour How to solve the knight's tour Traveling salesman problem Nearest neighbour algorithm Trees Tree (graph theory) Terminology Root node, root (computing) Leaf node Child node, parent node Ply Tree structure Tree data.
Four color theorem - a computer and are impractical for humans to check alone. The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See experimental mathematics. The lack of mathematical elegance was another factor; to paraphrase comments of the time "a good mathematical proof is like a poem -- this is a telephone directory!" Table of contents showTocToggle("show","hide") 1 Formal statement in graph theory 2 Generalizations 3 Real world counter examples Formal statement in graph theory To formally state the theorem, it is easiest to rephrase it in graph.
Compiler optimization - these architectures were designed to be easy for compiler writers to target. Types of Optimizations Techniques in optimization can be broken up along various dimensions: ; scope of the optimization: The effect of a particular compiler optimization can affect anywhere from a single statement to an entire program. Some examples of optimization scopes: Peephole optimizations: Usually performed late in the compilation process, peephole optimizations examine at most a few instructions, transforming instructions into other less expensive ones, such as turning a multiplication of x by two into an addition of x with itself. Local optimizations: These operate on a single basic block, a piece of straight-line code. Basic blocks have many useful properties, particularly order of execution of instructions, that makes it possible to do many simple optimizations on them with.
Computational geometry - and integrated circuit design systems routinely handling tens and hundreds of million points the difference between N2 and N log N boils down to the difference between seconds and days of computation. Hence the emphasis on computational complexity in computational geometry. Some core algorithms: Convex hull. Delaunay triangulation and Polygon triangulation. Voronoi diagram. Smallest bounding sphere. Closest pair of points. Line segment intersection. Minimal convex decomposition Boolean operations of polytopes. Ray casting (also known as Ray tracing). Some computational geometry problems: The museum problem. If a museum (represented by a polygon in the plane) wants to post guards (which see in all directions) to avoid getting robbed by a crook that could in principle drop from the ceiling, it is sufficient to post a guard at each vertex. This follows from.
Colouring algorithm - streams. Each of these may be overridden, changed or removed by using an input stream. The problem with this lies in creating "enough" colouring to make all of this function. This is sometimes called "bootstrapping." A variant of the coloring algorithm is known as the graph coloring algorithm. It is used in various allocation strategies, for example in register allocation in compiler development, and resource allocation in operating systems.
Sim - consisting of six dots ('vertices'). Each dot is connected to each other with a line. Players alternate coloring any uncolored line in their own color. Players try to avoid making triangles of their color; the player who completes a triangle of their color loses immediately. (A triangle is three dots, each connected to the other two with lines of the same color.) The other player is the winner. A simple theorem of Ramsey theory shows that no game of Sim can end in a tie; one player must lose by the end. Specifically, since R(3,3;2)=6, any coloring of the complete graph on 6 vertices must contain a monochromatic triangle, and therefore is not a tied position..
Ramsey's theorem - It states for any pair of positive integers (r,s) there exists an integer R(r,s) such that for any complete graph on R(r,s) vertices whose edges are coloured red or blue, there exists either a monochromatic complete subgraph on r vertices which is entirely blue or a monochromatic complete subgraph on s vertices which is entirely red. An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colors c, and any given integers n1,...,nc, there is a number, R(n1, ..., nc; c), such that if the edges of a complete graph of order R(n1, ..., nc; c) are colored with c different colors, then for some i between 1 and c, it must contain a.
Probabilistic method - probability is zero can be used to prove the non-existence of such an object). Another way to use the probabilistic method is by calculating the expected value of some random variable. If it can be shown that the random variable can take on a value less than the expected value, this proves that the random variable can also take on some value greater than the expected value. Example: One example of how we can use this method is to create a lower bound on the Ramsey number R(r,r;2). Suppose we have a complete graph on n vertices. We wish to show (for small enough values of n) that it is possible to color the edges of the graph in two colors (say red and blue) so that there is no complete.
List of mathematical topics (G-I) - algorithm -- Gentzen, Gerhard -- Germ (mathematics) -- Genus -- Geodesic -- Geodesic dome -- Geodesic flow -- Geographic coordinate system -- Geometer -- Geometers -- Geometric algebra -- Geometric Brownian motion -- Geometric distribution -- Geometric isomerism -- Geometric kite -- Geometric mean -- Geometric primitive -- Geometric progression -- Geometric series -- Geometric shape -- Geometric solid -- Geometry -- Geometry of numbers -- Gergonne point -- Germain -- Germain, Sophie -- Gibbard-Satterthwaite theorem -- Gibbs-Helmholtz equation -- Gibbs' phase rule -- Gibbs phenomenon -- Gift wrapping algorithm -- Gimel function -- GIMPS -- Gini coefficient -- Girsanov's theorem -- Girth -- Global field -- Globe -- Glome -- Glossary of field theory -- Glossary of graph theory -- Glossary of group theory -- Glossary of ring theory --.
Interval graph - Interval graph In graph theory, an interval graph is a graph that captures the intersections among a set of intervals on the real line. Formally, let be a set of intervals. Then the corresponding interval graph is G = (V, E) where and That is, the nodes of the graph are the intervals and there is an edge corresponding to each pair of intersecting intervals. Interval graphs are useful in modeling resource allocation problems in operations research. Each interval represents a request for a resource for a specific period of time. External Link Interval graph -- from Mathworld.
Homeomorphism (graph theory) - Homeomorphism (graph theory) A homeomorphism in graph theory is the relationship between two graphs G and G' such that if graph G' is obtained from subdivisions of one or more vertices of graph G, G' is said to be homeomorphic to G. This article is a stub. You can help Wikipedia by fixing it..
Hypergraph - In mathematics, the concept of hypergraph generalizes the notion of a graph. Informally a hypergraph is a graph whose edges, instead of each connecting just two vertices, connect more than two vertices. For any set X, let X[m] be the set of subsets of X whose elements have precisely m members. Then an m-hypergraph consists of a set X of vertices and a subset E of X[m]. A hypergraph is then defined to be a pair of sets (X,E) such that the pair constitute an m-hypergraph for some m. Commonly X is finite but this need not always be the case. It is also possible to define a hypergraph in a more general way as a pair of sets (X,E) where E is some arbitrary subset of the power set of.
Glossary of graph theory - Glossary of graph theory A loop in a graph or a digraph is an edge e in E whose endpoints are the same vertex. A digraph or graph is called simple if there are no loops and there is at most one edge between any pair of vertices. The example graph pictured to the right is a simple graph with vertex set V = {1, 2, 3, 4, 5, 6} and edge set E = (with the map w being the identity). An edge connects two vertices; these two vertices are said to be incident to the edge. The valency (or degree) of a vertex is the number of edges incident to it, with loops being counted twice. In the example graph vertices 1 and 3 have.
Graph - Graph A graph is In linguistics, a letter or symbol such as an alphabetic letter, a Chinese character, or a hieroglyph. In mathematics, the word graph can have two meanings. In graph theory, a graph is an object consisting of vertices (or nodes) and edges (or arcs) between pairs of vertices. The graph of a function f : X -> Y is the set of all pairs (x,f(x)) The graph of a relation, a generalisation of the graph of a function. This is a disambiguation page; that is, one that just points to other pages that might otherwise have the same name. If you followed a link here, you might want to go back and fix that link to point to the appropriate specific page..
Graph of a function - Graph of a function Mathematically, the graph of a function is the collection of all pairs (x, f(x)) of the function. Graphing is sometimes referred to as curve sketching. The graph of the function is {(1,a), (2,d), (3,c)}. The graph of the cubic polynomial on the real line is {(x,x3-9x) : x is a real number}. If the set is plotted on a Cartesian plane, the result is Therefore the graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis. The concept of the graph of a.
Graph reduction machine - Graph reduction machine A graph reduction machine is a special-purpose computer built to perform combinator calculations by graph reduction. Examples include the SKIM ("S-K-I machine") computer, built at the University of Cambridge, and the multiprocessor GRIP ("Graph Reduction In Parallel") computer, built at University College London. Table of contents showTocToggle("show","hide") 1 References 2 Related articles 3.
Graph paper - Graph paper Graph paper is paper that is printed with fine lines making up a grid. It is typically used for drawing diagrams, with the lines being used as guides for the drawing. It is commonly found in engineering settings for quick drawings and sketches. It can also be used for plotting mathematical functions manually. Some specialized forms of graph paper have logarithmic scales or allow plotting in polar coordinates. Quad paper is a common form of graph paper in imperial unit countries where the grid is a quarter inch apart printed in light blue and right to the edge of the paper. 3D graph paper is also available, but fairly rare. It uses a series of three guidelines forming a 60-degree grid, so that the.
Graph (mathematics) - Graph (mathematics) This article is about graphs in graph theory. See graph of a function and graph of a relation for other uses of "graph" in mathematics. In mathematics and computer science, a graph is a generalization of the simple concept of a set of dots, called vertices or nodes, connected by links, edges or arcs. "Nodes" and "arcs" are old notation. Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, i.e. numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) we have a directed graph. Structures that can be represented as graphs are.
