Iterated clique graphs and bordered compact surfaces

The clique graph K ( G ) of a graph G is the intersection graph of all its (maximal) cliques. A graph G is said to be K -divergent if the sequence of orders of its iterated clique graphs | K n ( G ) | tends to infinity with n , otherwise it is K -convergent. K -divergence is not known to be computable and there is even a graph on 8 vertices whose K -behavior is unknown. It has been shown that a regular Whitney triangulation of a closed surface is K -divergent if and only if the Euler characteristic of the surface is non-negative. Following this remarkable result, we explore here the existence of K -convergent and K -divergent (Whitney) triangulations of compact surfaces and find out that they do exist in all cases except (perhaps) where previously existing conjectures apply: it was conjectured that there is no K -divergent triangulation of the disk, and that there are no K -convergent triangulations of the sphere, the projective plane, the torus and the Klein bottle. Our results seem to suggest that the topology still determines the K -behavior in these cases.