Good sequencings for small Mendelsohn triple systems

TL;DR

This paper investigates specific orderings of points in small Mendelsohn triple systems to avoid certain local patterns, contributing new insights into their sequencing properties.

Contribution

It introduces the concept of ll-good sequencing for small Mendelsohn triple systems and analyzes their existence and properties.

Findings

Identifies conditions for ll-good sequencing in small MTS(v)

Provides constructions or classifications for small cases

Establishes bounds or non-existence results for certain parameters

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 $j-k+1 \leq \ell$.