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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 verticesB  1   ∈  B    andB  2   ∈  B    are adjacent if and only if|    B  1   ∩   B  2    |  >  0    . The  i  ‐block‐intersection graph  ( i ‐BIG) of   D   is the graph that has   B   as its vertex set, where two verticesB  1   ∈  B    andB  2   ∈  B    are adjacent if and only if|    B  1   ∩   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 classes in arbitrary TTSs, and then construct larger TTSs with both cyclic 2‐intersecting Gray codes and parallel classes.

Twofold triple systems with cyclic 2‐intersecting Gray codes