Hamiltonicity and restricted block-intersection graphs of -designs

Given a combinatorial design D with block set B , its traditional block-intersection graph G D is the graph having vertex set B such that two vertices b 1 and b 2 are adjacent if and only if b 1 and b 2 have non-empty intersection. In this paper, we consider the S -block-intersection graph, in which two vertices b 1 and b 2 are adjacent if and only if | b 1 ∩ b 2 | ∈ 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 ) } .