The Clique Behavior of Circulants With Three Small Jumps

Abstract

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 \(K^n(G)\) 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 \(C_n(a, b, c)\) with \(0 < a < b < c < \frac{n}{3}\). 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 \(C_n(1, 2, 4)\), which is clique divergent but no previously known technique could be used to prove it.