Pointed Conflict-Free Colouring of Digraphs

In a pointed confl ict-free partial (PCFP) colouring of a digraph, each vertex has at least one in-neighbour with unique colour. In this paper, it is proved that PCFP k-colourability of digraphs is NP-complete, for any k > 0. Nevertheless for paths and cycles, one can in linear time nd a PCFP colouring with a minimum number of colours and for a given tree, one can nd a PCFP 2-colouring. In this paper a bipartite digraph whose arcs start from the same part is called a one-way bipartite digraph. It is proved every one-way bipartite planar digraph has a PCFP 6-colouring, every one-way bipartite planar digraph whose each vertex has in-degree zero or greater than one, has a PCFP 5-colouring and every one-way bipartite planar digraph whose each vertex has in-degree zero or greater than two, has a PCFP 2-colouring. Two simple algorithms are proposed for nding a PCFP colouring of a given digraph such that the number of colours used is not more than the maximum out-degree of the vertices. For a digraph with a given PCFP colouring, it is shown how to recolour the vertices after vertex or arc insertion or deletion to obtain a PCFP colouring for the new digraph.