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Tuza's Conjecture
Tuza's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning triangles in undirected graphs. Statement In any graph G, one can define two quantities \nu(G) and \tau(G) based on the triangles in G. The quantity \nu(G) is the "triangle packing number", the largest number of edge-disjoint triangles that it is possible to find in G. It can be computed in polynomial time as a special case of the matroid parity problem. The quantity \tau(G) is the size of the smallest "triangle-hitting set", a set of edges that touches at least one edge from each triangle. Clearly, \nu(G)\le\tau(G)\le 3\nu(G). For the first inequality, \nu(G)\le\tau(G), any triangle-hitting set must include at least one edge from each triangle of the optimal packing, and none of these edges can be shared between two or more of these triangles because the triangles are disjoint. For the second inequality, \tau(G)\le 3\nu(G), one can construct a triangle-hitting set of size 3\nu(G) by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K5 Triangle Packing And Covering
K5, K05 or K-5 may be: Places * Gasherbrum I, the 11th highest mountain peak in the world * K-5 (Kansas highway), a state highway in Kansas * K5 Plan, vast defensive belt along the Cambodian-Thai border Transportation * Wings of Alaska, IATA airline designator * Kinner K-5, a light general and sport aircraft engine Vehicles * , a Royal Navy submarine sunk in 1921 * or , a 1940 British Royal Navy then Free French Navy * , a 1914 United States Navy K-class submarine * PRR K5, a 1929 American experimental 4-6-2 "Pacific" type steam locomotive * LNER Class K5, a class of British steam locomotives * GSR Class K5, an 1894 Irish steam locomotive * Chevrolet K5 Blazer, a 1969-91 full size SUV * Kia Optima, a car branded as K5 in some markets Weaponry * Daewoo Precision Industries K5, a pistol used by the South Korean military * Krupp K5, a railway gun of World War II Germany * Kaliningrad K-5, a Soviet-era air-to-air missile * K-5 (SLBM), a submarine-launched ballistic missile ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Treewidth
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests A forest is an ecosystem characterized by a dense community of trees. Hundreds of definitions of forest are used throughout the world, incorporating factors such as tree density, tree height, land use, legal standing, and ecological functio .... An example of graphs with treewidth at most 2 are the series–parallel graphs. The maximal graphs with treewidth exactly are called '' -trees'', and the graphs with treewidth at most are called '' partial -trees''. Many other well-studied graph families also have bounded treewidth. Treewidth may be formally defined in several equivalent ways: in terms of the size of the largest vertex set in a tree decomposition of the graph, in terms of the size of the largest clique in a chordal completi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Removal Lemma
In graph theory, the graph removal lemma states that when a graph contains few copies of a given subgraph, then all of the copies can be eliminated by removing a small number of edges. The special case in which the subgraph is a triangle is known as the triangle removal lemma. The graph removal lemma can be used to prove Roth's theorem on 3-term arithmetic progressions, and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem. It also has applications to property testing. Formulation Let H be a graph with h vertices. The graph removal lemma states that for any \epsilon > 0, there exists a constant \delta = \delta(\epsilon, H) > 0 such that for any n-vertex graph G with fewer than \delta n^h subgraphs isomorphic to H, it is possible to eliminate all copies of H by removing at most \epsilon n^2 edges from G. An alternative way to state this is to say that for any n-vertex graph G with o(n^h) subgraphs isomorphic to H, it is possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
