Bipartite Ramsey number pairs that involve combinations of cycles and odd paths

Abstract

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2$, $\dots ,G_k)$ is the least positive integer b, so that any coloring of the edges of $K_{b,b}$ with k colors, will result in a copy of $G_i$ in the ith color, for some i. 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 $K_{a^{\prime },b^{\prime }}$, with $a^{\prime }\ge b^{\prime }$, either a blue copy of $G_1$ exists or a red copy of $G_2$ exists if and only if $a^{\prime }\ge a$ and $b^{\prime }\ge b$. In [4], Faudree and Schelp considered bipartite Ramsey number pairs involving paths. Recently, Joubert, Hattingh and Henning showed, in [7] and [8], that $b_p(C_{2s},C_{2s})=(2s,2s-1)$ and $b(P_{2s},C_{2s})=2s-1$, for sufficiently large positive integers s. In this paper we will focus our attention on finding exact values for bipartite Ramsey number pairs that involve cycles and odd paths. Specifically, let s and r be sufficiently large positive integers. We will prove that $b_p(C_{2s},P_{2r+1})=(s+r,s+r-1)$ if $r\ge s+1$, $b_p(P_{2s+1},C_{2r})=(s+r,s+r)$ if $r=s+1$, and $b_p(P_{2s+1},C_{2r})=(s+r-1,s+r-1)$ if $r\ge s+2$.