Given a graph and a colouring of its vertices, a rainbow path is a path such that all its internal nodes are coloured distinctly. A graph is rainbow vertex-connected if between every pair of vertices there exists a rainbow path. We study the problem of deciding whether a graph can be coloured using k colours such that it is rainbow vertex-connected. Heggernes et al. (MFCS, 2018) conjectured that if every induced subgraph in G has a dominating diametral path, then G can always be rainbow coloured with diam ( G ) − 1 colours. We confirm their conjecture for chordal, bipartite and claw-free diametral path graphs. We complement these results by showing the conjecture does not hold without the condition on every induced subgraph. In this case, even though diam ( G ) colours are enough, it is NP-complete to determine whether a graph with a dominating diametral path of length three can be rainbow coloured with two colours.

Given a graph and a colouring of its vertices, a rainbow vertex path is a path between two vertices such that all the internal nodes of the path are coloured distinctly. A graph is rainbow vertex-connected if between every pair of vertices in the graph there exists a rainbow vertex path. We study the problem of deciding whether a given graph can be coloured using k or less colours such that it is rainbow vertex-connected. Note that every graph G needs at least diam(G)-1 colours to be rainbow vertex connected. Heggernes et al. [MFCS, 2018] conjectured that if G is a graph in which every induced subgraph has a dominating diametral path, then G can always be rainbow vertex coloured with diam(G)-1 many colours. In this work, we confirm their conjecture for chordal, bipartite and claw-free diametral path graphs. We complement these results by showing the conjecture does not hold if the condition on every induced subgraph is dropped. In fact we show that, in this case, even though diam(G) many colours are always enough, it is NP-complete to determine whether a graph with a dominating diametral path of length three can be rainbow vertex coloured with two colours.