A sharp threshold for rainbow connection in small-world networks

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc.G/, is the smallest number of colors that are needed in order to makeG rainbow connected. We prove that p D p lnn=n is a sharp threshold function for the property rc.S.n;p;H// 2 in the small-world networks. As by-products, our extension of the concept of independence in graph theory and generalized small-world network models are of independent interest. 2000 Mathematics Subject Classification: 05C82; 05C15; 05C40

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