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Hamiltonian Decomposition
In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs. In the undirected case a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that each factor is connected. Necessary conditions For a Hamiltonian decomposition to exist in an undirected graph, the graph must be connected and regular of even degree. A directed graph with such a decomposition must be strongly connected and all vertices must have the same in-degree and out-degree as each other, but this degree does not need to be even. Special classes of graphs Complete graphs Every complete graph with an odd number n of vertices has a Hamiltonian decomposition. This result, which is a special case of the Oberwolfach problem of decomposing complete graphs into isomorphic 2-factors, was attributed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walecki Decomposition (''powiat wałecki''), an administrative division in NW Poland
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Walecki or Wałecki may refer to: * Henryk Walecki, Polish inter-war communist activist *Wałcz County __NOTOC__ Wałcz County () is a unit of territorial administration and local government (powiat) in West Pomeranian Voivodeship, north-western Poland. It came into being on January 1, 1999, as a result of the Polish local government reforms passed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniela Kühn
Daniela Kühn (born 1973) is a German mathematician and the Mason Professor in Mathematics at the University of Birmingham in Birmingham, England.Staff profile University of Birmingham School of Mathematics, accessed 2012-09-12. She is known for her research in , and particularly in extremal combinatorics and . Biography Kühn earned the[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Petersen Graph
In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins. Definition and notation In Watkins' notation, ''G''(''n'', ''k'') is a graph with vertex set :\ and edge set :\ where subscripts are to be read modulo ''n'' and ''k'' < ''n''/2. Some authors use the notation ''GPG''(''n'', ''k''). Coxeter's notation for the same graph would be + , a combination of the Schläfli symbols for the regular ''n''-gon and star polygon from which the graph is formed. The Petersen graph itself is ''G''(5, 2) or + . Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one other edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Ladder
In graph theory, the Möbius ladder , for even numbers , is formed from an by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of (the utility graph ), has exactly four-cycles which link together by their shared edges to form a topological Möbius strip. Möbius ladders were named and first studied by . Properties For every even , the Möbius ladder is a nonplanar apex graph, meaning that it cannot be drawn without crossings in the plane but removing one vertex allows the remaining graph to be drawn without crossings. These graphs have crossing number one, and can be embedded without crossings on a torus or projective plane. Thus, they are examples of toroidal graphs. explores embeddings of these graphs onto higher genus surfaces. Möbius ladders are vertex-transitive – they have symmetries taking any vertex to any other vertex – but (with the exceptions of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halin Graph
In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree (graph theory), tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the Euclidean plane, plane so none of its edges cross (this is called a planar embedding), and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.''Encyclopaedia of Mathematics'', first Supplementary volume, 1988, , p. 281, articl"Halin Graph" and references therein. Halin graphs are named after German mathematician Rudolf Halin, who studied them in 1971.. The cubic graph, cubic Halin graphs – the ones in which each vertex touches exactly three edges – had already been studied over a century earlier by Thomas Kirkman, Kirkman. Halin graphs are polyhedral graphs, meaning that every Halin graph can be used to form the vertices a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected Graph (discrete mathematics), graph is said to be -vertex-connected (or -connected) if it has more than Vertex (graph theory), vertices and remains Connectivity (graph theory), connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. In complete graphs, there is no subset whose removal would disconnect the graph. Some sources modify the definition of connectivity to handle this case, by defining it as the size of the smallest subset of vertices whose deletion results in either a disconnected graph or a single vertex. For this variation, the connectivity of a complete graph K_n is n-1. An equivalent definition is that a graph with at least two vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon Base (geometry), base, a second base which is a Translation (geometry), translated copy (rigidly moved without rotation) of the first, and other Face (geometry), faces, necessarily all parallelograms, joining corresponding sides of the two bases. All Cross section (geometry), cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements, Euclid's ''Elements''. Euclid defined the term in Book XI as "a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms". However, this definition has been criticized for not being specific enough in regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called . The number of vertices in equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. If n = 1, it is an isolated loop. Terminology There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or ''n''-gon are also often used. The term ''n''-cycle is sometimes used in other settings. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Properties A cycle graph is: * 2-edge colorable, if and only if it has an even n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product Of Graphs
In graph theory, the Cartesian product of graphs and is a graph such that: * the vertex set of is the Cartesian product ; and * two vertices and are adjacent in if and only if either ** and is adjacent to in , or ** and is adjacent to in . The Cartesian product of graphs is sometimes called the box product of graphs arary 1969 The operation is associative, as the graphs and are naturally isomorphic. The operation is commutative as an operation on isomorphism classes of graphs, and more strongly the graphs and are naturally isomorphic, but it is not commutative as an operation on labeled graphs. The notation has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs. The square symbol is intended to be an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges. Examples * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herschel Graph
In graph theory, a branch of mathematics, the Herschel graph is a bipartite graph, bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. It is named after British astronomer Alexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs (but not of this graph). Definition and properties The Herschel graph has three vertices of degree four (the three blue vertices aligned vertically in the center of the illustration) and eight vertices of degree three. Each two distinct degree-four vertices share two degree-three neighbors, forming a four-vertex cycle with these shared neighbors. There are three of these cycles, passing through six of the eight degree-three vertices (red in the illustration). Two more degree-three vertices (blue) do not participate in these four- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Medial Graph
In the mathematical discipline of graph theory, the medial graph of plane graph ''G'' is another graph ''M(G)'' that represents the adjacencies between edges in the faces of ''G''. Medial graphs were introduced in 1922 by Ernst Steinitz to study combinatorial properties of convex polyhedra, although the inverse construction was already used by Peter Tait in 1877 in his foundational study of knots and links. Formal definition Given a connected plane graph ''G'', its medial graph ''M(G)'' has * a vertex for each edge of ''G'' and * an edge between two vertices for each face of ''G'' in which their corresponding edges occur consecutively. The medial graph of a disconnected graph is the disjoint union of the medial graphs of each connected component. The definition of medial graph also extends without modification to graph embeddings on surfaces of higher genus. Properties * The medial graph of any plane graph is a 4-regular plane graph. * For any plane graph ''G'', the me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grinberg's Theorem
In graph theory, Grinberg's theorem is a necessary condition for a planar graph to contain a Hamiltonian cycle, based on the lengths of its face cycles. If a graph does not meet this condition, it is not Hamiltonian. The result has been widely used to prove that certain planar graphs constructed to have additional properties are not Hamiltonian; for instance it can prove non-Hamiltonicity of some counterexamples to Tait's conjecture that cubic polyhedral graphs are Hamiltonian. Grinberg's theorem is named after Latvian mathematician Emanuel Grinberg, who proved it in 1968. Formulation A planar graph is a graph that can be drawn without crossings in the Euclidean plane. If the points belonging to vertices and edges are removed from the plane, the connected components of the remaining points form polygons, called ''faces'', including an unbounded face extending to infinity. A face is a if its boundary is formed by a cycle of and of the graph drawing. A Hamiltonian cycle in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
