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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.

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In this paper we compare the brushing number of a graph with the zero-forcing number of its line graph. We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph. Using a similar construction, we also prove the conjecture that the zero-forcing number of a graph is no more than the zero-forcing number of its line graph; moreover we prove that the brushing number of a graph is no more than the brushing number of its line graph. All three bounds are shown to be tight.