The bipartite ramsey number of cycles - Tivane, Amukelani (My University, 2021)
The 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.
A dissertation submitted in fulfilment of the requirements for the degree of Master of Science to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2022
Keywords: Ramsey number; s-bipartite Ramsey number