Cycles in the block-intersection graph of pairwise balanced designs
type: Item
instance of: scholarly article
title: Cycles in the block-intersection graph of pairwise balanced designs
zbMATH Open document ID: 0813.05041
DOI: 10.1016/0012-365X(93)E0140-Y
author: Donovan R. Hare
published in: Discrete Mathematics
publication date: 1995-02-06T00:00:00Z
review text: Let \(K\) be a set of positive integers, and let \(\lambda\) and \(v\) be positive integers so that \(v\) is greater than the largest integer belonging to \(K\). A pairwise balanced design \(\text{PBD}(v,K, \lambda)\) is a collection of subsets (called blocks) of a \(v\)-set \(V\) such that every block has cardinality equal to an integer in \(K\) and every pair of distinct elements of \(V\) lies in precisely \(\lambda\) blocks. The block- intersection graph of a pairwise balanced design is obtained by letting the vertices correspond to the blocks and defining two vertices to be adjacent if and only if the corresponding blocks have non-empty intersection. A graph \(G\) is said to be edge-pancyclic if every edge of \(G\) is contained in a cycle of each length \(\ell\) satisfying \(3\leq\ell\leq n\), where \(n\) is the order of \(G\). The author proves that the block-intersection graph of any pairwise balanced design with \(\lambda= 1\) and no blocks having two or fewer elements is edge- pancyclic.
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 719461
zbMATH Keywords: block design
zbMATH Keywords: pairwise balanced design
zbMATH Keywords: block-intersection graph
zbMATH Keywords: edge- pancyclic
reviewed by: Brian Alspach
MaRDI profile type: Publication
cites work: Edge-pancyclic block-intersection graphs
cites work: Q3487379
cites work: Q4873796
cites work: Q3137967
Wikidata QID: Q127976873
description: scientific article; zbMATH DE number 719461