Hamiltonicity and restricted block-intersection graphs of t-designs

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