Twofold triple systems without 2-intersecting Gray codes

Abstract

Given a combinatorial design $$\mathcal {D}$$D with block set $$\mathcal {B}$$B, the block-intersection graph (BIG) of $$\mathcal {D}$$D is the graph having $$\mathcal {B}$$B as its vertex set, and in which two vertices $$B_{1} \in \mathcal {B}$$B1źB and $$B_{2} \in \mathcal {B} $$B2źB are adjacent if and only if $$|B_{1} \cap B_{2}| > 0$$|B1źB2|>0. The i-block-intersection graph (i-BIG) of $$\mathcal {D}$$D is the graph having $$\mathcal {B}$$B as its vertex set, and in which two vertices $$B_{1} \in \mathcal {B}$$B1źB and $$B_{2} \in \mathcal {B}$$B2źB are adjacent if and only if $$|B_{1} \cap B_{2}| = i$$|B1źB2|=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 $$v \equiv 0$$vź0 or 1 (modulo 3) such that $$v \geqslant 6$$vź6, except for $$v \in \{7,9,10\}$$vź{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 $$v \equiv 0$$vź0 or 1 (modulo 3) such that $$v \geqslant 13$$vź13, except possibly for $$v \in \{13,15,16,27,28,30,31,33,34,37 \}$$vź{13,15,16,27,28,30,31,33,34,37}.