65th SAMS Congress
06-08 December 2022
Stellenbosch University
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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.