Locally C6 graphs are clique divergent

Larrión, F.; Neumann-Lara, Víctor

doi:10.1016/s0012-365x(99)00233-2

Cited by 32 publications

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“…Proof. As usual, we assume m ≤ n. The smallest case is when (m, n) = (3, 4) and, since G 2,3 ∼ = C 6 , G 3,4 is a locally C 6 graph, hence K -divergent by [16]. Otherwise, G m,n is expansive since G m,n contains a coaffine subgraph which is isomorphic to G 1,2 + G m−1,n−2 (see Fig.…”

Section: Clique Divergencementioning

confidence: 99%

The clique operator on matching and chessboard graphs

Larrión

Pizaña

Villarroel-Flores

2009

Discrete Mathematics

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Given positive integers m, n, we consider the graphs G n and G m,n whose simplicial complexes of complete subgraphs are the well-known matching complex M n and chessboard complex M m,n . Those are the matching and chessboard graphs. We determine which matching and chessboard graphs are clique-Helly. If the parameters are small enough, we show that these graphs (even if not clique-Helly) are homotopy equivalent to their clique graphs. We determine the clique behavior of the chessboard graph G m,n in terms of m and n, and show that G m,n is clique-divergent if and only if it is not clique-Helly. We give partial results for the clique behavior of the matching graph G n .

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Section: Clique Divergencementioning

confidence: 99%

The clique operator on matching and chessboard graphs

Larrión

Pizaña

Villarroel-Flores

2009

Discrete Mathematics

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“…For the non-orientable case, we shall use some concepts and results form [4]: Let A and B be graphs. A triangular covering map from A to B is any local isomorphism p : A → B, i.e.…”

Section: Proofmentioning

confidence: 99%

“…In particular, if one of A and B is a Whitney triangulation so is the other. All fibers p −1 (b) have the same cardinality: the number of sheets of p. Galois triangular covers constitute an important particular case: If Γ is a group of automorphisms of A and d(a, γ(a)) ≥ 4 holds for every a ∈ A and γ ∈ Γ\{1}, we can take the quotient graph A/Γ and then the natural projection p : A → A/Γ is a triangular covering map ( [4], Lemma 3.1). Note that every non-trivial element in Γ is a 4-coaffination of A.…”

Section: Proofmentioning

confidence: 99%

“…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].…”

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confidence: 99%

“…Indirect methods are scarce but more fruitful: For instance, any graph with a clique divergent retract is clique divergent [9]. Also, in any finite triangular covering map [4], the cover and the base have the same K-behaviour (see 4.2 below). Another indirect approach is to prove something stronger than clique divergence: expansivity in [10], rank divergence in this work.…”

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confidence: 99%

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Graph relations, clique divergence and surface triangulations

Larrión¹,

Neumann-Lara²,

Pizaña

2005

Journal of Graph Theory

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Abstract. This work has two aims: First, we introduce a powerful technique for proving clique divergence when the graph satisfies a certain symmetry condition. Second, we prove that each closed surface admits a clique divergent triangulation. By definition, a graph is clique divergent if the orders of its iterated clique graphs tend to infinity, and the clique graph of a graph is the intersection graph of its maximal complete subgraphs.

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On the Iterated Biclique Operator

Groshaus¹,

Montero²

2012

J. Graph Theory

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A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graph of G, denoted by KB(G), is the intersection graph of the bicliques of G. We say that a graph G diverges (or converges or is periodic) under an operator F whenever trueprefixlimk→∞|V(Fk(G))|=∞ (trueprefixlimk→∞Fk(G)=Fm(G) for some m, or Fk(G)=Fk+s(G) for some k and s≥2, respectively). Given a graph G, the iterated biclique graph of G, denoted by KBk(G), is the graph obtained by applying the biclique operator k successive times to G. In this article, we study the iterated biclique graph of G. In particular, we classify the different behaviors of KBk(G) when the number of iterations k grows to infinity. That is, we prove that a graph either diverges or converges under the biclique operator. We give a forbidden structure characterization of convergent graphs, which yield a polynomial time algorithm to decide if a given graph diverges or converges. This is in sharp contrast with the situsation for the better known clique operator, where it is not even known if the corresponding problem is decidable. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 181–190, 2013

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Locally C6 graphs are clique divergent

Cited by 32 publications

References 0 publications

The clique operator on matching and chessboard graphs

The clique operator on matching and chessboard graphs

Graph relations, clique divergence and surface triangulations

On the Iterated Biclique Operator

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