Graph decompositions in projective geometries.

Let PG(Fqv) be the (v−1)‐dimensional projective space over Fq and let Γ be a simple graph of order qk−1q−1 for some k. A 2−(v,Γ,λ) design over Fq is a collection ℬ of graphs (blocks) isomorphic to Γ with the following properties: the vertex set of every block is a subspace of PG(Fqv); every two distinct points of PG(Fqv) are adjacent in exactly λ blocks. This new definition covers, in particular, the well‐known concept of a 2−(v,k,λ) design over Fq corresponding to the case that Γ is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of Γ‐decompositions over F2 or F3 for which Γ 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 Γ is complete and λ=1, that is, the Steiner 2‐designs over a finite field. Also, we briefly touch the new topic of near resolvable 2−(v,2,1) designs over Fq. 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 Γ‐decompositions over a finite field that can be obtained by suitably labeling the vertices of Γ with the elements of a Singer difference set. [ABSTRACT FROM AUTHOR]