Whitney triangulations, local girth and iterated clique graphs

We study the dynamical behaviour of surface triangulations under the iterated application of the clique graph operator k, which transforms each graph G into the intersection graph kG of its (maximal) cliques. A graph G is said to be k-divergent if the sequence of the orders of its iterated clique graphs |V (knG)| tends to in4nity with n. If this is not the case, then G is eventually k-periodic, or k-bounded: kG ∼= kG for some m¿n. The case in which G is the underlying graph of a regular triangulation of some closed surface has been previously studied under the additional (Whitney) hypothesis that every triangle of G is a face of the triangulation: if G is regular of degree d, it is known that G is k-bounded for d= 3 and k-divergent for d= 4; 5; 6. We will show that G is k-bounded for all d¿ 7, thus completing the study of the regular case. Our proof works in the more general setting of graphs with local girth at least 7. As a consequence we obtain also the k-boundedness of the underlying graph G of any triangulation of a compact surface (with or without border) provided that any triangle of G is a face of the triangulation and that the minimum degree of the interior vertices of G is at least 7. c © 2002 Published by Elsevier Science B.V.