Báo cáo toán học: "A generalization of generalized Paley graphs and new lower bounds for R(3, q)"

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:A generalization of generalized Paley graphs and new lower bounds for R(3, q). | A generalization of generalized Paley graphs and new lower bounds for R 3 q Kang Wu Wenlong Su South China Normal University Wuzhou University Guangzhou Guangdong 510631 China Wuzhou Guangxi 543002 China wukang12345@ ramsey8888@ Haipeng Luo1 Xiaodong Xu2 Guangxi Academy of Sciences Nanning Guangxi 530007 China 1 haipengluo@ 2 xxdmaths@ Submitted Dec 31 2009 Accepted Apr 22 2010 Published May 7 2010 Mathematics Subject Classifications 05C55 Abstract Generalized Paley graphs are cyclic graphs constructed from quadratic or higher residues of finite fields. Using this type of cyclic graphs to study the lower bounds for classical Ramsey numbers has high computing efficiency in both looking for parameter sets and computing clique numbers. We have found a new generalization of generalized Paley graphs . automorphism cyclic graphs also having the same advantages. In this paper we study the properties of the parameter sets of automorphism cyclic graphs and develop an algorithm to compute the order of the maximum independent set based on which we get new lower bounds for 8 classical Ramsey numbers R 3 22 131 R 3 23 137 R 3 25 154 R 3 28 173 R 3 29 184 R 3 30 i 190 R 3 31 199 R 3 32 214. Furthermore we also get R 5 23 521 based on R 3 22 131. These nine results above improve their corresponding best known lower bounds. 1 Lower bounds for Ramsey numbers and generalized Paley graphs Let q1 q2 . qm 3 be given integers with m 2. The classical Ramsey number R q1 . qm is the minimum positive integer n satisfying the following condition For an arbitrary coloring of the complete graph Kn with m colors there is always a complete subgraph Kqi for some 1 i m such that every edge of Kqi has the i-th color. The determination of Ramsey numbers is a very difficult problem in combinatorics 1 . Various methods have been designed to compute their bounds. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N25 1 When Greenwood and Gleason determined the exact values