Tivane, Amukelani2023-11-212023-11-212021https://hdl.handle.net/10539/37059A dissertation submitted in fulfilment of the requirements for the degree of Master of Science to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2022The Ramsey number R(H1, H2) of two graphs H1 and H2 is the smallest positive integer n for which every red-blue coloring of the complete graph Kn of order n results in a subgraph of Kn isomorphic to H1 all of whose edges are colored red (called a red H1), or a subgraph of Kn isomorphic to H2 all of whose edges are colored blue (called a blue H2). The s-bipartite Ramsey number bs(H1, H2) of two bipartite graphs H1 and H2 is the smallest positive integer t, with t ≥ s, such that every red-blue coloring of the complete bipartite graph Ks,t results in a red H1 or a blue H2. For s = t, the sbipartite Ramsey number is known as the bipartite Ramsey number, which we denote by b(H1, H2). In this dissertation we investigate b(C2m, C2n) and bs(C2m, C2n) for m ≥ 2 and n ≥ 2.enRamsey numbers-bipartite Ramsey numberDissertation