Existential closure of block intersection graphs of infinite designs having infinite block size

2011 ◽  
Vol 19 (4) ◽  
pp. 317-327 ◽  
Author(s):  
Daniel Horsley ◽  
David A. Pike ◽  
Asiyeh Sanaei
Keyword(s):  
Block Size ◽  
Intersection Graphs ◽  
Existential Closure ◽  
Block Intersection

Existential closure of block intersection graphs of infinite designs having finite block size and index

2011 ◽  
Vol 19 (2) ◽  
pp. 85-94 ◽  
Author(s):  
David A. Pike ◽  
Asiyeh Sanaei
Keyword(s):  
Block Size ◽  
Intersection Graphs ◽  
Existential Closure ◽  
Block Intersection

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.

scholarly journals Pancyclic BIBD block-intersection graphs

2004 ◽  
Vol 284 (1-3) ◽  
pp. 205-208 ◽  
Author(s):  
Aygul Mamut ◽  
David A Pike ◽  
Michael E Raines
Keyword(s):  
Intersection Graphs ◽  
Block Intersection

scholarly journals Edge-pancyclic block-intersection graphs

1991 ◽  
Vol 97 (1-3) ◽  
pp. 17-24 ◽  
Author(s):  
Brian Alspach ◽  
Donovan Hare
Keyword(s):  
Intersection Graphs ◽  
Block Intersection

scholarly journals The connectivity of the block-intersection graphs of designs

1993 ◽  
Vol 3 (1) ◽  
pp. 5-8 ◽  
Author(s):  
Donovan R. Hare ◽  
William Mc Cuaig
Keyword(s):  
Intersection Graphs ◽  
Block Intersection

scholarly journals Silver Block Intersection Graphs of Steiner 2-Designs

2012 ◽  
Vol 29 (4) ◽  
pp. 735-746
Author(s):  
A. Ahadi ◽  
Nazli Besharati ◽  
E. S. Mahmoodian ◽  
M. Mortezaeefar
Keyword(s):  
Intersection Graphs ◽  
Block Intersection

Hamiltonicity and cycle extensions in 0-block-intersection graphs of balanced incomplete block designs

2015 ◽  
Vol 80 (3) ◽  
pp. 421-433 ◽  
Author(s):  
Jason T. LeGrow ◽  
David A. Pike ◽  
Jonathan Poulin
Keyword(s):  
Block Designs ◽  
Incomplete Block ◽  
Intersection Graphs ◽  
Balanced Incomplete Block Designs ◽  
Block Intersection ◽  
Balanced Incomplete Block

Cycle Extensions in BIBD Block-Intersection Graphs

2012 ◽  
Vol 21 (7) ◽  
pp. 303-310 ◽  
Author(s):  
Atif A. Abueida ◽  
David A. Pike
Keyword(s):  
Intersection Graphs ◽  
Block Intersection

Hamilton cycles in block-intersection graphs of triple systems

1999 ◽  
Vol 7 (4) ◽  
pp. 243-246 ◽  
Author(s):  
Peter Hor�k ◽  
David A. Pike ◽  
Michael E Raines
Keyword(s):  
Hamilton Cycles ◽  
Intersection Graphs ◽  
Triple Systems ◽  
Block Intersection

Hamilton cycles in restricted block-intersection graphs

2011 ◽  
Vol 61 (3) ◽  
pp. 345-353 ◽  
Author(s):  
Andrew T. Jesso ◽  
David A. Pike ◽  
Nabil Shalaby
Keyword(s):  
Hamilton Cycles ◽  
Intersection Graphs ◽  
Block Intersection
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