Graph Decompositions In Projective Geometries

Let PG(Fqv) be the (v-1)-dimensional projective space over Fq and let Gamma be a simple graph of order qk-1q-1 for some k. A 2-(v,Gamma,lambda) design over Fq is a collection beta of graphs (blocks) isomorphic to Gamma 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 lambda blocks. This new definition covers, in particular, the well-known concept of a 2-(v,k,lambda) design over Fq corresponding to the case that Gamma is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of Gamma-decompositions over F2 or F3 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, 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 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.