Rainbow K-Connection in Dense Graphs

An edge-coloured path is rainbow if the colours of its edges are distinct. For a positive integer k, an edge-colouring of a graph G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The rainbow k-connection number rck(G) is dened to be the minimum integer t such that there exists an edge-colouring of G with t colours which is rainbow k-connected. We consider rc2(G) when G has xed vertex-connectivity. We also consider rck(G) for large complete bipartite and multipartite graphs G with equipartitions. Finally, we determine sharp threshold functions for the properties rck(G) = 2 and rck(G) = 3, where G is a random graph. Related open problems are posed. AMS Subject Classiciation Numbers: 05C15, 05C40, 05C80