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Search: MSC category 05B05 ( Block designs [See also 51E05, 62K10] )

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1. CMB 2013 (vol 57 pp. 72)

Grari, A.
 Un Anneau Commutatif associÃ© Ã  un design symÃ©trique Dans les articles \cite{1}, \cite{2} et \cite{3}; l'auteur dÃ©veloppe une reprÃ©sentation d'un plan projectif fini par un anneau commutatif unitaire dont les propriÃ©tÃ©s algÃ©briques dÃ©pendent de la structure gÃ©omÃ©trique du plan. Dans l'article \cite{4}; il Ã©tend cette reprÃ©sentation aux designs symÃ©triques. Cependant l'auteur de l'article \cite{7} fait remarquer que la multiplication dÃ©finie dans ce cas ne peut Ãªtre associative que si le design est un plan projectif. Dans ce papier on mÃ¨nera une Ã©tude de cette reprÃ©sentation dans le cas des designs symÃ©triques. On y montrera comment on peut faire associer un anneau commutatif unitaire Ã  tout design symÃ©trique , on y prÃ©cisera certaines de ses propriÃ©tÃ©s, en particulier, celles qui relÃ¨vent de son invariance. On caractÃ©risera aussi les gÃ©omÃ©tries projectives finies de dimension supÃ©rieure moyennant cette reprÃ©sentation. Keywords:projective planes, symmetric designs, commutative ringsCategories:05B05, 16S99

2. CMB 2007 (vol 50 pp. 504)

Dukes, Peter; Ling, Alan C. H.
 Asymptotic Existence of Resolvable Graph Designs Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A \emph{block design} $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda$ elements of $\cA$. More generally, for a fixed simple graph $G$, a \emph{graph design} $\GD(v,G,\lambda)$ is a collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda$ elements of $\cA$. A famous result of Wilson says that for a fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all sufficiently large $v$ satisfying certain necessary conditions. A block (graph) design as above is \emph{resolvable} if $\cA$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs. Keywords:graph decomposition, resolvable designsCategories:05B05, 05C70, 05B10
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