Stellenbosch University
Bipartite Ramsey number pairs involving
cycles
Johannes H. Hattingh, University of
North Carolina, USA
Ernst J. Joubert\(^*\), University of
Johannesburg
SAMS Subject Classification Number: 11, 21
Let \(a\) and \(b\) be positive integers with \(a\geq b\). For bipartite graphs \(G_1\) and \(G_2\), the bipartite Ramsey number pair \((a,b)\), denoted by \(b_p(G_1,G_2)=(a,b)\), is an ordered pair of integers such that for any blue-red coloring of the edges of \(K_{r,t}\), with \(r\geq t\), either a blue copy of \(G_1\) exists, or a red copy of \(G_2\) exists, if and only if \(r\geq a\) and \(t\geq b\). In [1], Faudree and Schelp determined bipartite Ramsey number pairs for paths. In this paper we will focus on bipartite Ramsey number pairs that involve cycles. In particular, we will show, for \(s\) sufficiently large, that \(b_p(C_{2s},C_{2s})=(2s,2s-1).\)
References
[1] A. Gyàrfàs, C. C. Rousseau, R. H. Schelp, An extremal problem for paths in bipartite graphs. J. of Graph Theory. 8 (1984) 83–95.
