Good sequencings for small Mendelsohn triple systems

Abstract

A Mendelsohn triple system of order $v$ (or MTS$(v)$) is a decomposition of the complete graph into directed 3-cyles. We denote the directed 3-cycle with edges $(x,y)$, $(y,z)$ and $(z,x)$ by $(x,y,z)$, $(y,z,x)$ or $(z,x,y)$. An $\ell$-good sequencing of a MTS$(v)$ 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 $i < j < k$ and $k-i+1 \leq \ell$; or with $j < k < i$ and $i-j+1 \leq \ell$; or with $k < i < j$ and $j-k+1 \leq \ell$.