On clique divergent graphs with linear growth

Graph relations, clique divergence and surface triangulations

“…Here we construct some examples. Using the theory of clockwork graphs [5], we will be able to describe explicitly all their iterated clique graphs. In fact, our examples constitute the simplest and more symmetric kind of clockwork graphs.…”

“…It follows from [5] (see also the short version in [8]) that there is an isomorphism ψ n : K(R n 2m ) −→ R n+1 2m , but for the reader's convenience we shall explicitly describe the cliques of R n 2m and the isomorphism ψ n . For each vertex v = x u l ∈ R n 2m there is a clique Q u l which starts at this vertex: If l ∈ {0, 1}, the vertex v lies in the crown and the clique contains the crown's edge e = {x u l , x u+1 l } starting at v and the core's segment in the square containing e, that is, Q u l = {x u l , x u+1 l , x u 2 , x u 3 , ..., x u n+2 }.…”

“…The iterated clique graphs K n (G) are given by K n+1 (G) = K(K n (G)). We refer to [12,5,15] for the literature on iterated clique graphs. We study the dynamical behaviour of graphs under the iterates of the clique operator K. There are two main types of K-behaviour: G is clique convergent if K n (G) ∼ = K m (G) for some pair n < m; in other words, G is clique convergent iff G is, in the obvious sense, eventually clique periodic.…”

“…This has only been possible for a handful of cases: octahedra [9], locally C 6 graphs [4] and clockwork graphs [5]. Indirect methods are scarce but more fruitful: For instance, any graph with a clique divergent retract is clique divergent [9].…”

“…Theorem 4.3 is a strong result which will illustrate the power of our methods in §2, but in order to apply Theorem 2.6 to the proof of Theorem 4.3 we will need to have some rank divergent graphs. These shall be constructed in §3 using results on clockwork graphs from [5].…”