Twofold triple systems without 2-intersecting Gray codes

Abstract

Given a combinatorial design $\mathcal {D}$ with block set $\mathcal {B}$, the block-intersection graph (BIG) of $\mathcal {D}$ is the graph having $\mathcal {B}$ as its vertex set, and in which 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 having $\mathcal {B}$ as its vertex set, and in which 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 we present several constructions which together enable us to determine the complete spectrum of twofold triple systems with connected non-Hamiltonian 2-BIGs (equivalently, the complete spectrum of twofold triple systems that have no cyclic 2-intersecting Gray codes but for which the 2-BIGs are nevertheless connected); this spectrum consists of all orders $v \equiv 0$ or 1 (modulo 3) such that $v \geqslant 6$, except for $v \in \{7,9,10\}$. We also determine all but a finite number of the elements of the spectrum for twofold triple systems for which the 2-BIGs are connected but have no Hamilton path (i.e., for systems which lack 2-intersecting Gray codes but nevertheless have connected 2-BIGs); specifically, the spectrum is found to consist of every order $v \equiv 0$ or 1 (modulo 3) such that $v \geqslant 13$, except possibly for $v \in \{13,15,16,27,28,30,31,33,34,37 \}$.