Rainbow vertex k-connection in graphs

Let k be a positive integer and G be a k-connected graph. An edge-coloured path is rainbow if its edges have distinct colours. The rainbow k-connection number of G, denoted by rc(k)(G), is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths. The function rc(k)(G) was first introduced by Chartrand, Johns, McKeon, and Zhang in 2009, and has since attracted considerable interest. In this paper, we consider a version of the function rc(k)(G) which involves vertex-colourings. A vertex-coloured path is vertex-rainbow if its internal vertices have distinct colours. The rainbow vertex k-connection number of G, denoted by rvc(k)(G), is the minimum number of colours required to colour the vertices of G so that any two vertices of G are connected by k internally vertex-disjoint vertex-rainbow paths. We shall study the function rvc(k)(G) when G is a cycle, a wheel, and a complete multipartite graph. We also construct graphs G where rc(k)(G) is much larger than rvc(k) (G) and vice versa so that we cannot in general bound one of rc(k)(G) and ruc(k)(G) in terms of the other. (C) 2013 Elsevier B.V. All rights reserved.