Some Classes of Extended Directed Triple Systems and Numbers of Common Blocks

Abstract

An extended directed triple system of order \(v\) with an idempotent element (EDTS(\(v, a\))) is a collection of triples of the type \([x, y, z]\), \([x, y, x]\) or \((x, x, x)\) chosen from a \(v\)-set, such that every ordered pair (not necessarily distinct) belongs to only one triple and there are \(a\) triples of the type \((x, x, x)\). If such a design with parameters \(v\) and \(a\) exists, then it will have \(b_{v,a}\) blocks, where \(b_{v,a} = (v^2 + 2a)/3\). A necessary and sufficient condition for the existence of EDTS(\(v, 0\)) and EDTS(\(v, 1\)) are \(v \equiv 0 \pmod{3}\) and \(v \not\equiv 0 \pmod{3}\), respectively. In this paper, we have constructed two EDTS(\(v, a\))’s such that the number of common triples is in the set \(\{0, 1, 2, \ldots, b_{v,a} – 2, b_{v,a}\}\), for \(a = 0, 1\).