Iterated clique graphs with increasing diameters

  • Authors:

  • Claudson F. Bornstein_aff1n2;Jayme L. Szwarcfiter

  • Affiliations:

  • af2 Universidade Federal do Rio de Janeiro, Instituto de Matemática, NCE and COPPE, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brazil;af2 Universidade Federal do Rio de Janeiro, Instituto de Matemática, NCE and COPPE, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brazil

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 1998

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Abstract

A simple argument by Hedman shows that the diameter of a clique graph G differs by at most one from that of K(G), its clique graph. Hedman described examples of a graph G such that diam(K(G)) = diam(G) + 1 and asked in general about the existence of graphs such that diam(Ki(G)) = diam(G) + i. Examples satisfying this equality for i = 2 have been described by Peyrat, Rall, and Slater and independently by Balakrishnan and Paulraja. The authors of the former work also solved the case i = 3 and i = 4 and conjectured that such graphs exist for every positive integer i. The cases i ≥ 5 remained open. In the present article, we prove their conjecture. For each positive integer i, we describe a family of graphs G such that diam(Ki(G)) = diam(G) + i. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 147–154, 1998