Hamiltonicity and restricted block-intersection graphs of tt-designs

Given a combinatorial design DD with block set BB, its traditional block-intersection graph GDGD is the graph having vertex set BB such that two vertices b1b1 and b2b2 are adjacent if and only if b1b1 and b2b2 have non-empty intersection. In this paper, we consider the SS-block-intersection graph, in which two vertices b1b1 and b2b2 are adjacent if and only if |b1∩b2|∈S|b1∩b2|∈S. As our main result, we prove that {1,2,…,t−1}{1,2,…,t−1}-block-intersection graphs of tt-designs with parameters (v,t+1,λ)(v,t+1,λ) are Hamiltonian whenever t⩾3t⩾3 and v⩾t+3v⩾t+3, except possibly when (v,t)∈{(8,5),(7,4),(7,3),(6,3)}(v,t)∈{(8,5),(7,4),(7,3),(6,3)}.