Twofold triple systems with cyclic 2‐intersecting Gray codes

Abstract

Given a combinatorial design D with block set B, the block-intersection graph (BIG) of D is the graph that has B as its vertex set, where two vertices B1 ∈ B and B2 ∈ B are adjacent if and only if |B1 ∩ B2| > 0. The i-block-intersection graph (i-BIG) of D is the graph that has B as its vertex set, where two vertices B1 ∈ B and B2 ∈ B are adjacent if and only if |B1 ∩B2| = 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.

Decomposable Twofold Triple Systems with Non-Hamiltonian 2-Block Intersection Graphs

“…Block intersection graphs of designs have been a focus of attention since Ron Graham pondered whether STS block intersection graphs might be Hamiltonian, which was subsequently confirmed [4,24]. For some additional results concerning Hamilton cycles and similar properties in block intersection graphs of designs, see [1,3,5,9,12,15,16,22,23,26,28,30,31,35,36].…”

“…A Hamilton cycle in the 2-BIG of a TTS is equivalent to a cyclic Gray code [12], which leads to applications in coding theory. It is known that for v 4 such that v ≡ 0, 1 (mod 3) and v = 6, there exists a TTS(v) whose 2-BIG is Hamiltonian [12,16]. There also exists a TTS(v) whose 2-BIG is connected but non-Hamiltonian when v = 6 or v 12 and v ≡ 0, 1 (mod 3) [15].…”

Decomposable twofold triple systems with non-Hamiltonian 2-block intersection graphs

“…A Hamilton cycle in the 2-BIG of a TTS is equivalent to a cyclic Gray code [6], which leads to applications in coding theory. It is known that for v 4 such that v ≡ 0, 1 (mod 3) and v = 6, there exists a TTS(v) whose 2-BIG is Hamiltonian [6,10]. There also exists a TTS(v) whose 2-BIG is connected but non-Hamiltonian when v = 6 or v 12 and v ≡ 0, 1 (mod 3) [9].…”