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Tait's Conjecture
In mathematics, Tait's conjecture states that "Every K-vertex-connected graph, 3-connected Planar graph, planar cubic graph has a Hamiltonian cycle (along the edges) through all its Vertex (geometry), vertices". It was proposed by and disproved by , who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by . The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian. The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to ... [...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] |
Tutte Graph
In the mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte. It has chromatic number 3, chromatic index 3, girth 4 and diameter 8. The Tutte graph is a cubic polyhedral graph, but is non-hamiltonian. Therefore, it is a counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian cycle. Published by Tutte in 1946, it is the first counterexample constructed for this conjecture. Other counterexamples were found later, in many cases based on Grinberg's theorem. Construction From a small planar graph called the Tutte fragment, W. T. Tutte constructed a non-Hamiltonian polyhedron, by putting together three such fragments. The "compulsory" edges of the fragments, that must be part of any Hamiltonian path through the fragment, are connected at the central vertex; because any cycle can use only two of these three edges, there can be no Hamiltonian cycle. The resulting graph is 3-connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disproved Conjectures
A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition. In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and from other previously established theorems. The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof. In some areas of epistemology and theology, the notion of justification plays approximately the role of proof, while in jurisprudence the corresponding term is evidence, with "burden of proof" as a concept common to bot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Magazine
The ''Philosophical Magazine'' is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798;John Burnett"Tilloch, Alexander (1759–1825)" Dictionary of National Biography#Oxford Dictionary of National Biography, Oxford Dictionary of National Biography, Oxford University Press, Sept 2004; online edn, May 2006, accessed 17 Feb 2010 in 1822 Richard Taylor (editor), Richard Taylor became joint editor and it has been published continuously by Taylor & Francis ever since. Early history The name of the journal dates from a period when "natural philosophy" embraced all aspects of science. The very first paper published in the journal carried the title "Account of Mr Cartwright's Patent Steam Engine". Other articles in the first volume include "Methods of discovering whether Wine has been adulterated with any Metals prejudicial to Health" and "Description of the Apparatus used by Lavoisier to produce Water from its component Parts, Oxyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Graph
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the Vertex (geometry), vertices and Edge (geometry), edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the k-vertex-connected graph, 3-vertex-connected, planar graphs. Characterization The Schlegel diagram of a convex polyhedron represents its vertices and edges as points and line segments in the Euclidean plane, forming a subdivision of an outer convex polygon into smaller convex polygons (a convex drawing of the graph of the polyhedron). It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem, it is a k-vertex-connected graph, 3-vertex-connected graph. According to Steinitz's theorem, these two graph-theoretic properties are enough to completely Characterization (mathematics), characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs. That ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnette's Conjecture
Barnette's conjecture is an conjecture, unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite graph, bipartite polyhedral graph with cubic graph, three edges per vertex has a Hamiltonian cycle. Definitions A planar graph is an undirected graph that can be graph embedding, embedded into the Euclidean plane without any crossing number (graph theory), crossings. A planar graph is called polyhedral graph, polyhedral if and only if it is k-vertex-connected graph, 3-vertex-connected, that is, if there do not exist two vertices the removal of which would disconnect the rest of the graph. A graph is bipartite graph, bipartite if its vertices can be graph coloring, colored with two different colors such that each edge has one endpoint of each color. A graph is cubic graph, cubic (or regular graph, 3-regular) if each v ... [...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] |
Pentagonal Prism
In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a '' truncated pentagonal hosohedron'', represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product ×. The dual of a pentagonal prism is a pentagonal bipyramid. The symmetry group of a right pentagonal prism is ''D5h'' of order 20. The rotation group is ''D5'' of order 10. Volume The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal prism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinitz' Theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), 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 a 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 addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Connectivity
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two vertices and are called connected if contains a path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length (that is, they are the endpoints of a single edge), the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. An undirected graph is therefore disconnected if there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
