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- W4288286827 abstract "Let PG$(mathbb{F}_q^v)$ be the $(v-1)$-dimensional projective space over $mathbb{F}_q$ and let $Gamma$ be a simple graph of order ${q^k-1over q-1}$ for some $k$. A 2$-(v,Gamma,lambda)$ design over $mathbb{F}_q$ is a collection $cal B$ of graphs (textit{blocks}) isomorphic to $Gamma$ with the following properties: the vertex set of every block is a subspace of PG$(mathbb{F}_q^v)$; every two distinct points of PG$(mathbb{F}_q^v)$ are adjacent in exactly $lambda$ blocks. This new definition covers, in particular, the well known concept of a 2$-(v,k,lambda)$ design over $mathbb{F}_q$ corresponding to the case that $Gamma$ is complete. In this work of a foundational nature we illustrate how difference methods allow us to get concrete non-trivial examples of $Gamma$-decompositions over $mathbb{F}_2$ or $mathbb{F}_3$ for which $Gamma$ is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that $Gamma$ is complete and $lambda=1$, i.e., the Steiner 2-designs over a finite field. Also, we briefly touch the new topic of near resolvable 2-$(v,2,1)$ designs over $mathbb{F}_q$. This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of $Gamma$-decompositions over a finite field that can be obtained by suitably labeling the vertices of $Gamma$ with the elements of a Singer difference set." @default.
- W4288286827 created "2022-07-28" @default.
- W4288286827 creator A5025685163 @default.
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- W4288286827 date "2019-07-06" @default.
- W4288286827 modified "2023-09-24" @default.
- W4288286827 title "Graph decompositions in projective geometries" @default.
- W4288286827 doi "https://doi.org/10.48550/arxiv.1907.03194" @default.
- W4288286827 hasPublicationYear "2019" @default.
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