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bipartit
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There are 8 letters in BIPARTIT ( A1B3I1P3R1T1 )
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BIPARTIT - In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent ...
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In the mathematical field of graph theory, a Bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets * * * * U * * * {\displaystyle U} * and * * * * V * * * {\displaystyle V} * such that every edge connects a vertex in * * * * U * * * {\displaystyle U} * to one in * * * * V * * * {\displaystyle V} * . Vertex sets * * * * U * * * {\displaystyle U} * and * * * * V * * * {\displaystyle V} * are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.The two sets * * * * U * * * {\displaystyle U} * and * * * * V * * * {\displaystyle V} * may be thought of as a coloring of the graph with two colors: if one colors all nodes in * * * * U * * * {\displaystyle U} * blue, and all nodes in * * * * V * * * {\displaystyle V} * green, each edge has endpoints of differing colors, as is required in the graph coloring problem. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another green, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. * One often writes * * * * G * = * ( * U * , * V * , * E * ) * * * {\displaystyle G=(U,V,E)} * to denote a bipartite graph whose partition has the parts * * * * U * * * {\displaystyle U} * and * * * * V * * * {\displaystyle V} * , with * * * * E * * * {\displaystyle E} * denoting the edges of the graph. If a bipartite graph is not connected, it may have more than one bipartition; in this case, the * * * * ( * U * , * V * , * E * ) * * * {\displaystyle (U,V,E)} * notation is helpful in specifying one particular bipartition that may be of importance in an application. If * * * * * | * * U * * | * * = * * | * * V * * | * * * * {\displaystyle |U|=|V|} * , that is, if the two subsets have equal cardinality, then * * * * G * * * {\displaystyle G} * is called a balanced bipartite graph. If all vertices on the same side of the bipartition have the same degree, then * * * * G * * * {\displaystyle G} * is called biregular. |