scholarly journals Block intersection numbers of block designs

1980 ◽  
Vol 56 (2) ◽  
pp. 75-76
Author(s):  
Mitsuo Yoshizawa
Keyword(s):  
Block Designs ◽  
Intersection Numbers ◽  
Block Intersection

scholarly journals On some parametric classifications of quasi-symmetric 2-designs

2015 ◽  
Vol 46 (3) ◽  
pp. 269-280
Author(s):  
Debashis Ghosh ◽  
Lakshmi Kanta Dey
Keyword(s):  
Intersection Numbers ◽  
Block Intersection

Quasi-symmetric $2$-designs with block intersection numbers $x$ and $y$, where $y=x+4$ and $x > 0$ are considered. If $D(v, b, r, k, \lambda; x, y)$ is a quasi-symmetric $2$-design with above condition, then it is shown that the number of such designs is finite, whenever $3\leq x \leq 68$. Moreover, the non-existence of triangle free quasi-symmetric $2$-designs under these parameters is obtained.

Comparison of the Bounds for the Block Intersection Number of Incomplete Block Designs

1977 ◽  
Vol 26 (1-4) ◽  
pp. 53-60
Author(s):  
Sanpei Kageyama
Keyword(s):  
Intersection Number ◽  
Block Designs ◽  
Incomplete Block ◽  
General Bound ◽  
Block Intersection ◽  
Balanced Incomplete Block ◽  
Block Intersection Number

Firstly, we give a complete comparison between Agrawal's bounds (1964a) and Shah's bounds (1965) given for the block intersection number of incomplete block designs. That is, it is asserted that Agrawal's bound is generally superior to that given by Shah for certain partially balanced incomplete bloCk (PBIB) designs validating Shah's bounds. Secondly, we compare Agrawal's bound with another general bound for the block intersection numer.

scholarly journals Existentially Closed BIBD Block-Intersection Graphs

10.37236/988 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Neil A. McKay ◽  
David A. Pike
Keyword(s):  
Block Size ◽  
Combinatorial Design ◽  
Block Designs ◽  
Intersection Graphs ◽  
Triple Systems ◽  
Empty Intersection ◽  
Existentially Closed ◽  
Vertex Set ◽  
Block Intersection ◽  
Balanced Incomplete Block

A graph $G$ with vertex set $V$ is said to be $n$-existentially closed if, for every $S \subset V$ with $|S|=n$ and every $T \subseteq S$, there exists a vertex $x \in V-S$ such that $x$ is adjacent to each vertex of $T$ but is adjacent to no vertex of $S-T$. Given a combinatorial design ${\cal D}$ with block set ${\cal B}$, its block-intersection graph $G_{{\cal D}}$ is the graph having vertex set ${\cal B}$ 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 study BIBDs (balanced incomplete block designs) and when their block-intersection graphs are $n$-existentially closed. We characterise the BIBDs with block size $k \geq 3$ and index $\lambda=1$ that have 2-e.c. block-intersection graphs and establish bounds on the parameters of BIBDs with index $\lambda=1$ that are $n$-e.c. where $n \geq 3$. For $\lambda \geq 2$ and $n \geq 2$, we prove that only simple $\lambda$-fold designs can have $n$-e.c. block-intersection graphs. In the case of $\lambda$-fold triple systems we show that $n \geq 3$ is impossible, and we determine which 2-fold triple systems (i.e., BIBDs with $k=3$ and $\lambda=2$) have 2-e.c. block-intersection graphs.

Sign in / Sign up

Export Citation Format

Share Document