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

For bipartite graphs G 1 , G 2 , ..., G k , the bipartite Ramsey number b(G1, ( G 1 , G 2 , ..., Gk) 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 Gi i in the i th color, for some i . For bipartite graphs G 1 and G 2 , the bipartite Ramsey number pair ( a, b ) , denoted by bp(G1, 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 2 s , C 2 s ) = ( 2 s, 2s s -1) 1 ) and b(P2s, ( P 2 s , C 2 s ) = 2s s - 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 2 s , P 2 r + 1 ) = ( s + r, s + r -1) 1 ) if r > s + 1, b p ( P 2 s + 1 , C 2 r ) = ( s + r, s + r) ) if r = s + 1, and b p ( P 2 s + 1 , C 2 r ) = ( s + r - 1, , s + r - 1) ) if r > s + 2. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.