Shonda GOSSELIN

(University of Winnipeg, Canada)

Vertex-transitive q-complementary uniform hypergraphs

A k-uniform hypergraph $(V,E)$ is (cyclically) q-complementary if there is a permutation $\theta\in Sym(V)$ such that the sets $E, E^{\theta}, E^{\theta2},\ldots, E^{\theta^{q-1}}$ partition the set of all k-subsets of $V$. The 2-complementary 2-uniform hypergraphs are the well studied self-complementary graphs. The vertex-transitive q-complementary k-uniform hypergraphs form examples of large sets of isomorphic designs which are point-transitive.

The well known Paley graphs are both vertex-transitive and self-complementary. These graphs have a high level of symmetry and many interesting properties. In this talk, we introduce Paley's construction and present some examples, and then we use Paley's algebraic technique to generalize this construction and find some `Paley-like' vertex-transitive q-complementary k-uniform hypergraphs, for $k\geq 2$ and q prime. This will establish necessary and sufficient conditions on the order of these structures in the case where q is prime, for certain values of the rank k. Group theoretic results due to Burnside and Zassenhaus imply that, for these values of k, every vertex-transitive q-complementary k-uniform hypergraph of prime order can be obtained from some Paley-like uniform hypergraph by a switching operation.

Serdecznie zapraszamy wszystkich chêtnych !