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Keywords:

  • cycle extension;
  • block design;
  • intersection graph;
  • polynomial time algorithm

Abstract

A cycle C in a graph G is extendable if there is some other cycle in G that contains each vertex of C plus one additional vertex. A graph is cycle extendable if every non-Hamilton cycle in the graph is extendable. A balanced incomplete block design, BIBDinline image, consists of a set V of v elements and a block set inline image of k-subsets of V such that each 2-subset of V occurs in exactly λ of the blocks of inline image. The block-intersection graph of a design inline image is the graph inline image having inline image as its vertex set and such that two vertices of inline image are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block-intersection graph of any BIBDinline image is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block-intersection graph.