## Abstract

For bipartite graphs G_{1},G_{2},…,G_{k}, the bipartite Ramsey number b(G_{1},G_{2}, …,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′,b′}, with a^{′}≥b^{′}, either a blue copy of G_{1} exists or a red copy of G_{2} exists if and only if a^{′}≥a and b^{′}≥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≥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≥s+2.

Original language | English |
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Article number | 114283 |

Journal | Discrete Mathematics |

Volume | 348 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2025 |

## Keywords

- Bipartite graph
- Cycle
- Path
- Ramsey

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics