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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 to W ... [...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 and graph theory. Biography Kühn earned the Certificate of Advanced Studies in Mathematics ([...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 and ) ... [...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 into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the plane so none of its edges cross (this is called 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 Halin graphs – the ones in which each vertex touches exactly three edges – had already been studied over a century earlier by Kirkman. Halin graphs are polyhedral graphs, meaning that every Halin graph can be used to form the vertices and edges of a convex polyhedron, and the polyhedra formed from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains 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. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with ''n'' vertices has connectivity ''n'' − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is ''k''-connected if, for every pair of its vertices, it is possible to find ''k'' vertex-independent paths connecting these vertices; see Menger's theorem . This definition produces the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All 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 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 relation to the nature of the bases, which caused confusion among later geometry writers. Oblique prism An oblique prism is a prism in which the joining edges and faces are ''not perpendicular'' to the bas ... [...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. 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 number of vertices * 2-regular * 2-v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product Of Graphs
Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to: Mathematics *Cartesian closed category, a closed category in category theory *Cartesian coordinate system, modern rectangular coordinate system *Cartesian diagram, a construction in category theory *Cartesian geometry, now more commonly called analytic geometry *Cartesian morphism, formalisation of ''pull-back'' operation in category theory * Cartesian oval, a curve *Cartesian product, a direct product of two sets * Cartesian product of graphs, a binary operation on graphs *Cartesian tree, a binary tree in computer science Philosophy * Cartesian anxiety, a hope that studying the world will give us unchangeable knowledge of ourselves and the world * Cartesian circle, a potential mistake in reasoning * Cartesian doubt, a form of methodical skepticism as a basis for philosophical rigor *Cartesian dualism, the philosophy of the distinction between mind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herschel Graph
In graph theory, a branch of mathematics, the Herschel graph is a 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 and eight vertices of degree three. Each of the three pairs of degree-four vertices form a cycle of four vertices, in which they alternate with two of the degree-three vertices. Each of the two remaining degree-three vertices is adjacent to a degree-three vertex from each of these three cycles. The Herschel graph is a polyhedral graph; this means that it is a planar graph, one that can be drawn in the plane with none of its edges crossin ... [...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 Graph may refer to: Mathematics * Graph (discrete mathematics), a structure made of vertices and edges ** Graph theory, the study of such graphs ... [...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] |
