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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: The disjoint m-flower intersection problem for latin squares. | The disjoint m-flower intersection problem for latin squares James G. Lefevre School of Mathematics and Physics University of Queensland Brisbane QLD 4072 Australia j gl@maths.uq.edu.au Thomas A. McCourt School of Mathematics and Physics University of Queensland Brisbane QLD 4072 Australia tom.mccourt@uqconnect.edu.au Submitted Sep 1 2010 Accepted Jan 18 2011 Published Feb 21 2011 Mathematics Subject Classification 05B15 Abstract An m-flower in a latin square is a set of m entries which share either a common row a common column or a common symbol but which are otherwise distinct. Two m-flowers are disjoint if they share no common row column or entry. In this paper we give a solution of the intersection problem for disjoint m-flowers in latin squares that is we determine precisely for which triples n m x there exists a pair of latin squares of order n whose intersection consists exactly of x disjoint m-flowers. 1 Introduction Intersection problems for latin squares were first considered by Fu 10 . Since then the area has been extensively investigated see 6 for a survey of results up until 1990. Subsequent results can be found in 7 8 1 3 and 9 . Intersection problems between pairs of Steiner triple systems were first considered by Lindner and Rosa 12 . Subsequently the intersection problem between pairs of Steiner triple systems V Vi and V V2 in which the intersection of V1 and V2 is composed of a number of isomorphic copies of some specified partial triple system have also been considered. Mullin Poplove and Zhu 15 considered the case where the partial triple system in question was a triangle. Furthermore Lindner and Hoffman 11 considered pairs of Steiner triple systems of order v intersecting in a v-21 -flower and some other THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P42 1 possibly empty set of triples Chang and Lo Faro 4 considered the same problem for Kirkman triple systems. In 5 Chee investigated the intersection problem for Steiner triple systems in which
