A new family of expansive graphs

  • Authors:

  • Martín Matamala;José Zamora

  • Affiliations:

  • Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile and Centro de Modelamiento Mat ...;Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

An affine graph is a pair (G,@s) where G is a graph and @s is an automorphism assigning to each vertex of G one of its neighbors. On one hand, we obtain a structural decomposition of any affine graph (G,@s) in terms of the orbits of @s. On the other hand, we establish a relation between certain colorings of a graph G and the intersection graph of its cliques K(G). By using the results we construct new examples of expansive graphs. The expansive graphs were introduced by Neumann-Lara in 1981 as a stronger notion of the K-divergent graphs. A graph G is K-divergent if the sequence |V(K^n(G))| tends to infinity with n, where K^n^+^1(G) is defined by K^n^+^1(G)=K(K^n(G)) for n=1. In particular, our constructions show that for any k=2, the complement of the Cartesian product C^k, where C is the cycle of length 2k+1, is K-divergent.