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 $v\equiv 0(\mathrm{mod}3)$ and $v\not\equiv 0(\mathrm{mod}3)$, respectively. In this paper, we have constructed two EDTS($v,a$)’s such that the number of common triples is in the set $\{0,1,2,\dots ,b_{v,a}–2,b_{v,a}\}$, for $a=0,1$.