# Block-avoiding point sequencings of directed triple systems

## TL;DR

This paper investigates the existence of special point orderings in directed triple systems, proving conditions under which such sequences exist or do not, and providing computational analysis for small cases.

## Contribution

It establishes the existence and non-existence of v-good sequencings in DTS(v) for all relevant v, advancing understanding of their combinatorial structure.

## Key findings

- Existence of v-good sequencing for all v ≡ 0,1 mod 3.
- Non-existence of v-good sequencing for v ≥ 7, v ≡ 0,1 mod 3.
- Computational results for all nonisomorphic DTS(v) with v ≤ 7.

## Abstract

A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the complete directed graph $\vec{K_v}$ 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 $v \equiv 0,1 \bmod {3}$. Further, for all positive integers $v \equiv 0,1 \bmod {3}$, $v \geq 7$, 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 $v \leq 7$.