Perfect blocking sets in P3-designs

Abstract

A well-known problem in Design Theory is the study of the possible existence of blocking sets in Steiner systems. In this paper, we introduce the concept of perfect blocking sets in G-designs and determine all the possible v for which there exist P 3-designs having perfect blocking sets.

Blocking sets for cycles and paths designs

“…In general, when we have a blocking set T for a G-design Σ = (X, B) we might want that the elements of T are distributed in an optimal and homogeneous way in the blocks of B. So in [4] the following definition is given:…”

“…In [4] the spectrum of P 3 -designs having a perfect blocking set is determined. So it is proved that:…”

“…Theorem 18 ( [4]). If T is a perfect blocking set of any P 3 -design of order v, then c = 1, v is a square and…”

“…We study also perfect blocking sets, analizing the idea of a blocking set distributed in an optimal and homogeneous way. The notion was introduced in [4] and it requires that any block contains a constant number of edges between vertices of the blocking set T and vertices of C X (T ). In [4], the spectrum of P 3 -designs with a perfect blocking set is determined.…”

“…The notion was introduced in [4] and it requires that any block contains a constant number of edges between vertices of the blocking set T and vertices of C X (T ). In [4], the spectrum of P 3 -designs with a perfect blocking set is determined. In this paper, we easily determine the spectrum of C 4 -designs having a perfect blocking set, as it is follows from the result on the spectrum of largely blocked C 4 -designs.…”