# Twofold triple systems with cyclic 2-intersecting Gray codes

## TL;DR

This paper characterizes all twofold triple systems with Hamiltonian and Hamilton path 2-block-intersection graphs, providing a complete spectrum of such systems with cyclic 2-intersecting Gray codes.

## Contribution

It introduces new constructions for TTSs with Hamiltonian 2-BIGs and determines the complete spectrum of TTSs with these properties, including those with Hamilton paths.

## Key findings

- Complete spectrum of TTSs with Hamiltonian 2-BIGs established.
- Constructed larger TTSs with cyclic 2-intersecting Gray codes.
- Proved lower bounds on partial parallel classes in TTSs.

## Abstract

Given a combinatorial design $\mathcal{D}$ with block set $\mathcal{B}$, the block-intersection graph (BIG) of $\mathcal{D}$ is the graph that has $\mathcal{B}$ as its vertex set, where two vertices $B_{1} \in \mathcal{B}$ and $B_{2} \in \mathcal{B} $ are adjacent if and only if $|B_{1} \cap B_{2}| > 0$. The $i$-block-intersection graph ($i$-BIG) of $\mathcal{D}$ is the graph that has $\mathcal{B}$ as its vertex set, where two vertices $B_{1} \in \mathcal{B}$ and $B_{2} \in \mathcal{B}$ are adjacent if and only if $|B_{1} \cap B_{2}| = i$. In this paper several constructions are obtained that start with twofold triple systems (TTSs) with Hamiltonian $2$-BIGs and result in larger TTSs that also have Hamiltonian $2$-BIGs. These constructions collectively enable us to determine the complete spectrum of TTSs with Hamiltonian $2$-BIGs (equivalently TTSs with cyclic $2$-intersecting Gray codes) as well as the complete spectrum for TTSs with $2$-BIGs that have Hamilton paths (i.e., for TTSs with $2$-intersecting Gray codes). In order to prove these spectrum results, we sometimes require ingredient TTSs that have large partial parallel classes; we prove lower bounds on the sizes of partial parallel clasess in arbitrary TTSs, and then construct larger TTSs with both cyclic $2$-intersecting Gray codes and parallel classes.