Block-avoiding point sequencings of directed triple systems

Abstract

A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the complete directed graph $\overrightarrow{K_v}$ into transitive triples. A $v$-good sequencing of a DTS$(v)$ is a permutation of the points of the design, say $[x_1\cdots x_v]$, such that, for every triple $(x,y,z)$ in the design, it is not the case that $x=x_i$, $y=x_j$ and $z=x_k$ with $i<j<k$. We prove that there exists a DTS$(v)$ having a $v$-good sequencing for all positive integers $v\equiv 0,1\mathrm{mod}3$. Further, for all positive integers $v\equiv 0,1\mathrm{mod}3$, $v\ge 7$, we prove that there is a DTS$(v)$ that does not have a $v$-good sequencing. We also derive some computational results concerning $v$-good sequencings of all the nonisomorphic DTS$(v)$ for $v\le 7$.