Quasi-symmetric 3-designs with triangle-free graph

Abstract

A~TRACT. The following result is proved: Let D be a quasi-symmetric 3-design with intersection numbers x, y(0 ~< x < y < k). D has no three distinct blocks such that any two of them intersect in x points if and only if D is a Hadamard 3-design, or D has a parameter set (v, k, 2) where v = (2 + 2)(22 + 42 + 2) + 1, k = 22 + 32 + 2 and 2 = 1, 2 ..... or D is a complement of one of these designs.

Quasi‐symmetric designs with fixed difference of block intersection numbers

“…Triangle free quasisymmetric designs with x = 0 were first studied in Shrikhande [12]. Triangle free quasisymmetric 3-designs are completely classified in Pawale [9]. Recently in [10], Pawale characterize triangle free quasi-symmetric designs with non-zero intersection numbers and k = 2y − x, with trivial design.…”

Non Existence of Triangle Free Quasi-symmetric Designs

“…In [8], it is proved that 'if D is triangle free quasi-symmetric 3-design then D is a Hadamard 3-design, or D has a parameter set (v, k, λ) where v = (λ + 2)(λ 2 + 4λ + 2) + 1, k = λ 2 + 3λ + 2 and λ = 1, 2, . .…”

On quasi-symmetric designs with intersection difference three

“…So quasi-symmetric designs whose block graph and its complement are connected give rise to primitive strongly regular graphs. So earlier papers, such as [1,11,[18][19][20][21][22]25], and [23] may be viewed under this wider umbrella.…”