Cycle Extensions in BIBD Block-Intersection Graphs
Article first published online: 3 AUG 2012
© 2012 Wiley Periodicals, Inc.
Journal of Combinatorial Designs
Volume 21, Issue 7, pages 303–310, July 2013
How to Cite
Abueida, A. A. and Pike, D. A. (2013), Cycle Extensions in BIBD Block-Intersection Graphs. J. Combin. Designs, 21: 303–310. doi: 10.1002/jcd.21328
- Issue published online: 16 APR 2013
- Article first published online: 3 AUG 2012
- Manuscript Revised: 6 JUL 2012
- Manuscript Received: 12 MAR 2012
- cycle extension;
- block design;
- intersection graph;
- polynomial time algorithm
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, BIBD, consists of a set V of v elements and a block set of k-subsets of V such that each 2-subset of V occurs in exactly λ of the blocks of . The block-intersection graph of a design is the graph having as its vertex set and such that two vertices of are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block-intersection graph of any BIBD is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block-intersection graph.