On the complexity of rainbow vertex colouring diametral path graphs

Abstract

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 $\mathrm{diam}\left(G\right)-1$ 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 $\mathrm{diam}\left(G\right)$ 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.