On the clique behavior of circulants with three small jumps

The clique graph K(G) of a graph G is the intersection graph of all its (maximal) cliques, and G is said to be clique divergent if the order of its n-th iterated clique graph Kn(G) tends to infinity with n. In general, deciding whether a graph is clique divergent is not known to be computable. We characterize the dynamical behavior under the clique operator of circulant graphs of the form Cn(a, b, c) with 0 < a < b < c < n3 : Such a circulant is clique divergent if and only if it is not clique-Helly. Owing to the Dragan-Szwarcfiter Criterion to decide clique-Hellyness, our result implies that the clique divergence of these circulants can be decided in polynomial time. Our main difficulty was the case Cn(1, 2, 4), which is clique divergent but no previously known technique could be used to prove it.