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@ A. Pawel Wojda Faculty of Applied Mathemetics AGH University of Science and Technology Cracow Poland wojda@ 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 . 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 .
