Vertex-transitive self-complementary uniform hypergraphs

  • Authors:

  • Primo Potočnik;Mateja Šajna

  • Affiliations:

  • Institute of Mathematics, Physics, and Mechanics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia and Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 100 ...;Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON K1N 6N5, Canada

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2009

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Abstract

In this paper we examine the orders of vertex-transitive self-complementary uniform hypergraphs. In particular, we prove that if there exists a vertex-transitive self-complementary k-uniform hypergraph of order n, where k=2^@? or k=2^@?+1 and n=1(mod2^@?^+^1), then the highest power of any prime dividing n must be congruent to 1 modulo 2^@?^+^1. We show that this necessary condition is also sufficient in many cases-for example, for n a prime power, and for k=3 and n odd-thus generalizing the result on vertex-transitive self-complementary graphs of Rao and Muzychuk. We also give sufficient conditions for the existence of vertex-transitive self-complementary uniform hypergraphs in several other cases. Since vertex-transitive self-complementary uniform hypergraphs are equivalent to a certain kind of large sets of t-designs, the results of the paper imply the corresponding results in design theory.