Cycles in the block-intersection graph of pairwise balanced designs
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.
scientific article; zbMATH DE number 719461