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Cycle Double Cover
In graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide a double cover of the graph: each edge belongs to exactly two faces. It is an unsolved problem, posed by George Szekeres and Paul Seymour and known as the cycle double cover conjecture, whether every bridgeless graph has a cycle double cover. The conjecture can equivalently be formulated in terms of graph embeddings, and in that context is also known as the circular embedding conjecture. Formulation The usual formulation of the cycle double cover conjecture asks whether every bridgeless undirected graph has a collection of cycles such that each edge of the graph is contained in exactly two of the cycles. The requirement that the graph be bridgeless is an obvious necessary condition for such a set of cycles to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Double Cover
Petersen is a common Danish patronymic surname, meaning ''"son of Peter"''. There are other spellings. Petersen may refer to: People In arts and entertainment * Adolf Dahm-Petersen, Norwegian voice specialist * Anja Petersen, German operatic soprano and university lecturer * Anker Eli Petersen, Faroese writer and artist * Ann Petersen, Belgian actress * Chris Petersen (born 1963), American child actor * Devon Petersen (born 1986), South African darts player * Elmer Petersen, American artist * Gustaf Munch-Petersen, Danish writer and painter * Joel Petersen, bass guitarist * John Hahn-Petersen, Danish actor * Josef Petersen, Danish novelist * Patrick Petersen, American actor * Paul Petersen, American movie actor, singer, novelist, and activist * Robert E. Petersen, publisher, auto museum founder * Robert Storm Petersen, Danish cartoonist, writer, animator, illustrator, painter and humorist * Sandy Petersen, American game designer * Uwe Fahrenkrog-Petersen, German musician * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vizing's Theorem
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than the maximum degree of the graph. At least colors are always necessary, so the undirected graphs may be partitioned into two classes: "class one" graphs for which colors suffice, and "class two" graphs for which colors are necessary. A more general version of Vizing's theorem states that every undirected multigraph without loops can be colored with at most colors, where is the multiplicity of the multigraph. The theorem is named for Vadim G. Vizing who published it in 1964. Discovery The theorem discovered by Russian mathematician Vadim G. Vizing was published in 1964 when Vizing was working in Novosibirsk and became known as Vizing's theorem. Indian mathematician R. P. Gupta independently discovered the theorem, while undertaking his doctorate (1965-1967). Examples When , the graph must itself be a matching, with no two e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentBranko Grünbaum Hrvatska enciklopedija LZMK. and a professor at the in . He received his Ph.D. in 1957 from Hebrew University of Jerusalem< ...
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Nowhere-zero Flow
In graph theory, a nowhere-zero flow or NZ flow is a network flow that is nowhere zero. It is intimately connected (by duality) to coloring planar graphs. Definitions Let ''G'' = (''V'',''E'') be a digraph and let ''M'' be an abelian group. A map ''φ'': ''E'' → ''M'' is an ''M''-circulation if for every vertex ''v'' ∈ ''V'' :\sum_ \phi(e) = \sum_ \phi(e), where ''δ''+(''v'') denotes the set of edges out of ''v'' and ''δ''−(''v'') denotes the set of edges into ''v''. Sometimes, this condition is referred to as Kirchhoff's law. If ''φ''(''e'') ≠ 0 for every ''e'' ∈ ''E'', we call ''φ'' a nowhere-zero flow, an ''M''-flow, or an NZ-flow. If ''k'' is an integer and 0 < , ''φ''(''e''), < ''k'' then ''φ'' is a ''k''-flow. Other notions Let ''G'' = (''V'',''E'') be an . An ori ...[...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] |
Orientability
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Graph
In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross. Examples Any graph that can be embedded in a plane can also be embedded in a torus. A toroidal graph of genus 1 can be embedded in a torus but not in a plane. The Heawood graph, the complete graph K7 (and hence K5 and K6), the Petersen graph (and hence the complete bipartite graph K3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks, and all Möbius ladders are toroidal. More generally, any graph with crossing number 1 is toroidal. Some graphs with greater crossing numbers are also toroidal: the Möbius–Kantor graph, for example, has crossing number 4 and is toroidal. Properties Any toroidal graph has chromatic number at most 7. The complete graph K7 provides an example of a toroidal graph with chromatic number 7. Any triangle-free toroidal graph h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconnected Graph
In graph theory, a biconnected graph is a connected and "nonseparable" graph, meaning that if any one vertex were to be removed, the graph will remain connected. Therefore a biconnected graph has no articulation vertices. The property of being 2-connected is equivalent to biconnectivity, except that the complete graph of two vertices is usually not regarded as 2-connected. This property is especially useful in maintaining a graph with a two-fold redundancy, to prevent disconnection upon the removal of a single edge (or connection). The use of biconnected graphs is very important in the field of networking (see Network flow), because of this property of redundancy. Definition A biconnected undirected graph is a connected graph that is not broken into disconnected pieces by deleting any single vertex (and its incident edges). A biconnected directed graph is one such that for any two vertices ''v'' and ''w'' there are two directed paths from ''v'' to ''w'' which have no vert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell Complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces\emptyset = X_ \subset X_0 \subset X_1 \subset \cdotssuch that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to D^k, to X_ by continuous gluing maps g^k_\alpha: \partial e^k_\alpha \to X_. The maps are also called attaching maps. Each X_k is called the k-skeleton of the complex. The topology of X = \cup_ X ... [...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] |
Tietze's Graph
In the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz Friedrich Tietze, who showed in 1910 that the Möbius strip can be subdivided into six regions that all touch each other – three along the boundary of the strip and three along its center line – and therefore that graphs that are embedded onto the Möbius strip may require six colors. The boundary segments of the regions of Tietze's subdivision (including the segments along the boundary of the Möbius strip itself) form an embedding of Tietze's graph. Relation to Petersen graph Tietze's graph may be formed from the Petersen graph by replacing one of its vertices with a triangle. Like the Tietze graph, the Petersen graph forms the boundary of six mutually touching regions, but on the projective plane rather than on the Möbius strip. If one cuts a hole from this subdivision of the projective plane, surrounding a single verte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
