engleski

Graph decompositions in projective geometries

In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of ‐ decompositions over GF(2) or GF(3) for which 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 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 GF(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 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.

design over a finite field ; difference family ; difference set ; graph decomposition ; group divisible design over a finite field ; projective space ; spread

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