Four colours are needed for closed-neighbourhood conflict-free colouring of all planar graphs

Source abstract

A colouring of the vertices of a graph is conflict-free with respect to closed neighbourhoods if every closed neighbourhood contains a colour that appears on exactly one of its vertices. The colouring need not be proper. Every planar graph has such a colouring with four colours. Fabrici, Lužar, Rindošová and Soták conjectured that three colours always suffice. We disprove this conjecture. We give a simple gadget construction that turns every planar graph of chromatic number four into a planar graph with no such colouring with three colours. This also shows that it is NP-complete to decide whether three colours suffice for a planar graph. We also give a planar graph on $25$ vertices that needs four colours. Further, we give critical planar graphs on $25$ vertices with $52$ and with $51$ edges, an infinite family of plane triangulations that need four colours, and an infinite family of critical examples.