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 Bipartite Ramsey Numbers b(C2m; K2,2). | The Bipartite Ramsey Numbers b C2m K2 2 Zhang Rui Sun Yongqi School of Computer and Information Technology Beijing Jiaotong University Beijing P. R. China yqsun@bj Submitted Nov 26 2010 Accepted Feb 13 2011 Published Feb 28 2011 Mathematics Subject Classifications 05C55 05C38 Abstract Given bipartite graphs H1 and H2 the bipartite Ramsey number b H1 H2 is the smallest integer b such that any subgraph G of the complete bipartite graph Kbb either G contains a copy of H1 or its complement relative to Kbb contains a copy of H2. It is known that b K2 2 K2 2 5 b K2 3 K23 9 b K2 4 K2 4 14 and b K3 3 K3 3 17. In this paper we study the case H1 being even cycles and H2 being K2 2 prove that b C6 K2 2 5 and b C2m K2 2 m 1 for m 4. Keywords bipartite graph Ramsey number even cycle 1 Introduction We consider only finite undirected graphs without loops or multiple edges. For a graph G with vertex-set V G and edge-set E G we denote the order and the size of G by p G V G and q G E G . Ỏ G and A G are the minimum degree and the maximum degree of G respectively. Let Km n be a complete m by n bipartite graph that is Km n consists of m n vertices partitioned into sets of size m and n and the mn edges between them. Pk is a path on k vertices and Ck is a cycle of length k. Let H1 and H2 be bipartite graphs the bipartite Ramsey number b H1 H2 is the smallest integer b such that given any subgraph G of the complete bipartite graph Kb b either G contains a copy of Hl or there exists a copy of H2 in the complement of G relative to Kb b. Obviously we have b H1 H2 b H2 H1 . Beineke and Schwenk 1 showed that b K2 2 K2 2 5 b K2 4 K2A 13 b K3 3 K33 17. This research was supported by NSFC 60973011 60803034 and SRFDF 20090009120007 200801081017 . THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P51 1 In particular they proved that b K2 n K2 n 4n 3 for n odd and less than 100 except possibly n 59 or n 95. Carnielli and Carmelo 2 proved that b K2 n K2 n 4n 3 if 4n 3 is a prime power. They
