The s-bipartite Ramsey numbers involving K_(2,3) and K_(3,3)

摘要

A complete bipartite graph with the number of two partitions s and t is denoted by K_(s,t). For a positive integer s and two bipartite graphs G and H, the s-bipartite Ramsey number BR_s(G, H) of G and H is the smallest integer t such that every 2-coloring of the edges of K_(s,t) contains the a copy of G with the first color or a copy of H with the second color. In this paper, by using an integer linear program and the solver Gurobi Optimizer 8.0, we determine all the exact values of BR_s(K_(2,3), K(3,3)) for all possible s. More precisely, we show that BR_s(K_(2,3), K_(3,3)) = 13 for s e {8,9}, BR_s(K_(2,3),K_(3,3)) = 12 for s ∈ {10,11}, BR_s(K_(2,3),K_(3,3)) = 10 for s = 12, BR_s(K_(2,3),K_(3,3)) = 8 for s ∈ {13,14}, BR_s(K(2,3),K(3,3)) = 6 for s ∈ {15,16,…, 20}, and BR_s(K_(2,3),K(3,3)) = 4 for s > 21. This extends the results presented in [Zhenming Bi, Drake Olejniczak and Ping Zhang, "The s-Bipartite Ramsey Numbers of Graphs K_(2,3) and K(3,3)", Journal of Combinatorial Mathematics and Combinatorial Computing 106, (2018) 257-272].