TAILIEUCHUNG - Báo cáo toán học: "Vertex-transitive q-complementary uniform hypergraphs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Vertex-transitive q-complementary uniform hypergraphs. | Vertex-transitive q-complementary uniform hypergraphs Shonda Gosselin Department of Mathematics and Statistics University of Winnipeg 515 Portage Avenue Winnipeg MB R3B 2E9 Canada Submitted Feb 11 2010 Accepted Apr 28 2011 Published May 8 2011 Mathematics Subject Classification 05C65 05B05 05E20 05C85 Abstract For a positive integer q a k-uniform hypergraph X V E is q-complementary if there exists a permutation 0 on V such that the sets E Ee Ee . Eeq partition the set of k-subsets of V. The permutation 0 is called a q-antimorphism of X. The well studied self-complementary uniform hypergraphs are 2-complementary. For an integer n and a prime p let n p max i pi divides n . In this paper we prove that a vertex-transitive q-complementary k-hypergraph of order n exists if and only if nra p 1 mod q 1 for every prime number p in the case where q is prime k bq or k bq 1 for a positive integer b k and n 1 mod qi 1 . We also find necessary conditions on the order of these structures when they are t-fold-transitive and n t mod qi 1 for 1 t k in which case they correspond to large sets of isomorphic t-designs. Finally we use group theoretic results due to Burnside and Zassenhaus to determine the complete group of automorphisms and q-antimorphisms of these hypergraphs in the case where they have prime order and then use this information to write an algorithm to generate all of these objects. This work extends previous analagous results for vertex-transitive self-complementary uniform hypergraphs due to Muzychuk Potocnik Sajna and the author. These results also extend the previous work of Li and Praeger on decomposing the orbitals of a transitive permutation group. Key words Self-complementary hypergraph t-complementary hypergraph Uniform hypergraph Transitive hypergraph Complementing permutation Large set of t-designs Supported by a University of Winnipeg Major Research Grant THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P100 1 1 Introduction .

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