On the Complexity of Rainbow Vertex Colouring Diametral Path Graphs

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

An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study

On the Complexity of Role Colouring Planar Graphs, Trees and Cographs

We prove several results about the complexity of the role colouring problem. A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. We show that the problem of finding a role colouring with $1< k <n$ colours is NP-hard for planar graphs. We show that restricting the problem to trees yields a polynomially solvable case, as long as $k$ is either constant or has a constant difference with $n$, the number of vertices in the tree. Finally, we prove that cographs are always $k$-role-colourable for $1<k\leq n$ and construct such a colouring in polynomial time