Francisco Larrión

Locally C 6 graphs are clique divergent

The clique graph kG of a graph G is the intersection graph of the family of all maximal complete subgraphs of G. The iterated clique graphs knG are defined by k0G=G and kn+1G=kknG. A graph G is said to be k-divergent if V(knG) tends to infinity with n. A graph is locally C6 if the neighbours of any given vertex induce an hexagon. We prove that all locally C6 graphs are k-divergent and that the diameters of the iterated clique graphs also tend to infinity with n while the sizes of the cliques remain bounded.

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 (k n G)| tends to in4nity with n. If this is not the case, then G is eventually k-periodic, or k-bounded: k n G ∼ k m G 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.

A Family of Clique Divergent Graphs with Linear Growth

We present an infinite set A of finite graphs such that for any graph G e A the order | V(kn(G))| of the n-th iterated clique graph kn(G) is a linear function of n. We also give examples of graphs G such that | V(kn(G))| is a polynomial of any given positive degree.

Clique divergent graphs with unbounded sequence of diameters

Abstract The clique graph kG of a graph G is the intersection graph of the family of all maximal complete subgraphs of G . The iterated clique graphs k n G are defined by k 0 G = G and k n +1 G = kk n G . A graph G is said to be k -divergent if | V ( k n G )| tends to infinity with n . We provide examples of k -divergent graphs such that the diameters of the iterated clique graphs also tend to infinity with n . Furthermore, the sizes of the cliques and even the chromatic numbers remain bounded.

Clique divergent clockwork graphs and partial orders

S. Hazan and V. Neumann-Lara proved in 1996 that every finite partially ordered set whose comparability graph is clique null has the fixed point property and they asked whether there is a finite poset with the fixed point property whose comparability graph is clique divergent. In this work we answer that question by exhibiting such a finite poset. This is achieved by developing further the theory of clockwork graphs. We also show that there are polynomial time algorithms that recognize clockwork graphs and decide whether they are clique divergent.

On clique divergent graphs with linear growth

We study the dynamical behaviour of simple graphs under the iterated application of the clique graph operator k, which transforms each finite graph G into the intersection graph kG of its (maximal) cliques. The graph G is said to be clique divergent if the sequence of the orders o(knG) of the iterated clique graphs of G tends to infinity with n, and G is said to have linear growth if this divergent sequence is bounded by a linear function of n. In this work, we introduce an important family of graphs (the clockwork graphs) which is closed under the clique operator and contains clique divergent graphs with strictly linear growth, i.e., o(knG) = o(G) + rn, where r is any fixed positive integer. We apply our results to give examples of clique divergent graphs having non-strict linear growth.

On expansive graphs

The clique graph K(G) of a graph G, is the intersection graph of its (maximal) cliques, and G is K-divergent if the orders of its iterated clique graphs K(G),K^2(G),K^3(G),... tend to infinity. A coaffine graph has a symmetry that maps each vertex outside of its closed neighbourhood. For these graphs we study the notion of expansivity, which implies K-divergence.

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K^0(G)=G, K^n^+^1(G)=K(K^n(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that K^n(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G~K^n(G) for all n, moreover K^n(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.

The fundamental group of the clique graph

Given a finite connected bipartite graph B=(X,Y) we consider the simplicial complexes of complete subgraphs of the square B^2 of B and of its induced subgraphs B^2[X] and B^2[Y]. We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the clique graph K(G), the line graph L(G) and the total graph T(G).

Contractibility and the clique graph operator

To any graph G we can associate a simplicial complex @D(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever @D(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whose clique graph is a cone is contractible.