Cycle Extensions in BIBD Block-Intersection Graphs

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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, BIBDmath formula, consists of a set V of v elements and a block set math formula of k-subsets of V such that each 2-subset of V occurs in exactly λ of the blocks of math formula. The block-intersection graph of a design math formula is the graph math formula having math formula as its vertex set and such that two vertices of math formula are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block-intersection graph of any BIBDmath formula is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block-intersection graph.

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