Smallest defining sets of super-simple 2 - (v, 4,1) directed designs
Abstract
A
2−(v,k,λ)
directed design (or simply a
2−(v,k,λ)DD
) is super-simple if its underlying
2−(v,k,2λ)BIBD
is super-simple, that is, any two blocks of the
BIBD
intersect in at most two points. A
2−(v,k,λ)DD
is simple if its underlying
2−(v,k,2λ)BIBD
is simple, that is, it has no repeated blocks.
A set of blocks which is a subset of a unique
2−(v,k,λ)DD
is said to be a defining set of the directed design. A smallest defining set, is a defining set which has smallest cardinality. In this paper simultaneously we show that the necessary and sufficient condition for the existence of a super-simple
2−(v,4,1)DD
is
v≡1 (mod 3)
and for these values except
v=7
, there exists a super-simple
2−(v,4,1)DD
whose smallest defining sets have at least a half of the blocks. And also for all
ϵ>0
there exists
v
0
(ϵ)
such that for all admissible
v>
v
0
there exists a
2−(v,4,1)DD
whose smallest defining sets have at least
(5/8−
c
v
)∣B∣
blocks, for suitable positive constant c.