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Báo cáo toán học: "Cyclic partitions of complete 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: Cyclic partitions of complete uniform hypergraphs. | Cyclic partitions of complete uniform hypergraphs Artur Szymanski szymanski@artgraph.eu A. Pawel Wojda Faculty of Applied Mathemetics AGH University of Science and Technology Cracow Poland wojda@agh.edu.pl Submitted Jun 4 2010 Accepted Aug 5 2010 Published Sep 1 2010 Mathematics Subject Classifications 05C65 Abstract TAk By Kn we denote the complete k-uniform hypergraph of order n 1 k n 1 i.e. the hypergraph with the set Vn 1 2 . n of vertices and the set Ợ of edges. If there exists a permutation ơ of the set Vn such that E ơ E . ơq 1 E is a partition of the set V then we call it cyclic q-partition of Ka1 and ơ is said to be a q k -complementing. In the paper for arbitrary integers k q and n we give a necessary and sufficient condition for a permutation to be q k -complementing permutation of Ky. By Kn we denote the hypergraph with the set of vertices Vn and the set of edges 2Vn 0 Vn . If there is a permutation ơ of Vn and a set E c 2Vn 0 Vn such that E ơ E . ơp-1 E is a p-partition of 2Vn 0 Vn then we call it a cyclic p-partition of Kn and we say that ơ is p-complementing. We prove that Kn has a cyclic p-partition if and only if p is prime and n is a power of p and n p . Moreover any p-complementing permutation is cyclic. 1 Preliminaries and results Throughout the paper we will write Vn 1 . n . For a set X we denote by A the set of all k-subsets of X. A hypergraph H V E is said to be k-uniform if E c k the cardinality of any edge is equal to k . We shall always assume that the set of vertices V of a hypergraph of order n is equal to vn. The complete k-uniform hypergraph of order nA o kI OA-A All KaI 1a A T ỉ 1 In OKI oo ĨA1 - tty ÍV t 1 T oJ- KT Ino -A KAOKKK1A I I -a 1 A KAKI A I í dio CoC IV lol is denoted by JKn hence JKn Vn I k . Let Ơ be a peiinuLaLion or the set Vn let q be a positive integer and let E c p . If E ơ E a2 E . aq-1 E is a partition of V we call it a cyclic q-partition and Ơ is said to be q k -complementing. It is The research of APW was .