An edge-colouring of a graph G is rainbow k-connected if, for any two vertices of G, there are k internally vertex-disjoint paths joining them, each of which is rainbow (i.e., all edges of each path have distinct colours). The minimum number of colours for which there exists a rainbow k-connected colouring for G is the rainbow k-connection number of G, and is denoted by rck(G). The function rck(G) was introduced by Chartrand et al. in 2008, and has since attracted considerable interest. In this note, we shall consider the function rck(G) for complete bipartite and multipartite graphs, highly connected graphs, and random graphs.
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