Characterising clique convergence for locally cyclic graphs of minimum degree δ ≥ 6

The clique graph kG of a graph G has as its vertices the cliques (maximal complete subgraphs) of G, two of which are adjacent in kG if they have non-empty intersection in G. We say that G is clique convergent if knG≅kmG for some n≠m, and that G is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree δ≥6, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called “triangular-shaped graphs”.