Clique colourings of geometric graphs [article]

Colin McDiarmid, Dieter Mitsche, Pawel Pralat
2018 arXiv   pre-print
A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given n points x_1, ...,x_n in the plane, and a threshold r>0, the corresponding geometric graph has vertex set {v_1,...,v_n}, and distinct v_i and v_j are adjacent when the Euclidean distance between x_i and x_j is at most r. We investigate the clique chromatic number of such
more » ... . We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph RG in the plane, where n random points are independently and uniformly distributed in a suitable square. We see that as r increases from 0, with high probability the clique chromatic number is 1 for very small r, then 2 for small r, then at least 3 for larger r, and finally drops back to 2.
arXiv:1701.02693v2 fatcat:ajbg2da26vhs7lw2tgh2nm5ym4