R. Villarroel-Flores

On the structure of the h-vector of a paving matroid

We give two proofs that the h-vector of any paving matroid is a pure O-sequence, thus answering in the affirmative a conjecture made by Stanley, for this particular class of matroids. We also investigate the problem of obtaining good lower bounds for the number of bases of a paving matroid given its rank and number of elements.

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.

Posets, clique graphs and their homotopy type

To any finite poset P we associate two graphs which we denote by @W(P) and @?(P). Several standard constructions can be seen as @W(P) or @?(P) for suitable posets P, including the comparability graph of a poset, the clique graph of a graph and the 1-skeleton of a simplicial complex. We interpret graphs and posets as simplicial complexes using complete subgraphs and chains as simplices. Then we study and compare the homotopy types of @W(P), @?(P) and P. As our main application we obtain a theorem, stronger than those previously known, giving sufficient conditions for a graph to be homotopy equivalent to its clique graph. We also introduce a new graph operator H that preserves clique-Hellyness and dismantlability and is such that H(G) is homotopy equivalent to both its clique graph and the graph G.

Some split exact sequences in the cohomology of groups

Abstract We analyze the split exact sequences of (co)homology groups associated to the spaces of Dwyer which give rise to the centralizer decomposition and subgroup decomposition of the classifying space BG of a finite group. In the first instance these sequences have infinite length. We show that they give rise to finite sequences which are also split and exact. The sequences arise as the first page of a spectral sequence.

On the clique behavior of circulants with three small jumps

Abstract In the study reported in this extended abstract we characterize the clique behavior of circulant graphs of the form C n ( a , b , c ) with 0 a b c n 3 : Such a circulant is clique divergent if and only if it is not clique-Helly. The main difficulty found here, was the case C n ( 1 , 2 , 4 ) which is clique divergent, but no previously known technique could be used to prove it.

On self-clique shoal graphs

The clique graph of a graph G is the intersection graph K ( G ) of its (maximal) cliques, and G is self-clique if K ( G ) is isomorphic to? G . A graph G is locally H if the neighborhood of each vertex is isomorphic to? H . Assuming that each clique of the regular and self-clique graph? G is a triangle, it is known that G can only be r -regular for r ? { 4 , 5 , 6 } and G must be, depending on r , a locally H graph for some H ? { P 4 , P 2 ? P 3 , 3 P 2 } . The self-clique locally P 4 graphs are easy to classify, but only a family of locally? H self-clique graphs was known for H = P 2 ? P 3 , and another one for H = 3 P 2 .We study locally P 2 ? P 3 graphs (i.e.? shoal graphs). We show that all previously known shoal graphs were self-clique. We give a bijection from (finite) shoal graphs to 2-regular digraphs without directed 3-cycles. Under this translation, self-clique graphs correspond to self-dual digraphs, which simplifies constructions, calculations and proofs. We compute the numbers, for each n ? 28 , of self-clique and non-self-clique shoal graphs of order n , and also prove that these numbers grow at least exponentially with? n .

On self-clique graphs with triangular cliques

A graph is an { r , s } -graph if the set of degrees of their vertices is { r , s } . A clique of a graph is a maximal complete subgraph. The clique graph K ( G ) of a graph G is the intersection graph of all its cliques. A graph G is self-clique if G is isomorphic to K ( G ) . We show the existence of self-clique { 5 , 6 } -graphs whose cliques are all triangles, thus solving a problem posed by Chia and Ong (2012).

Iterated clique graphs and bordered compact surfaces

Abstract The clique graph K ( G ) of a graph G is the intersection graph of all its (maximal) cliques. A graph G is said to be K -divergent if the sequence of orders of its iterated clique graphs | K n ( G ) | tends to infinity with n , otherwise it is K -convergent . K -divergence is not known to be computable and there is even a graph on 8 vertices whose K -behavior is unknown. It has been shown that a regular Whitney triangulation of a closed surface is K -divergent if and only if the Euler characteristic of the surface is non-negative. Following this remarkable result, we explore here the existence of K -convergent and K -divergent (Whitney) triangulations of compact surfaces and find out that they do exist in all cases except (perhaps) where previously existing conjectures apply: it was conjectured that there is no K -divergent triangulation of the disk, and that there are no K -convergent triangulations of the sphere, the projective plane, the torus and the Klein bottle. Our results seem to suggest that the topology still determines the K -behavior in these cases.