Computational approaches to restricted and connected size Ramsey numbers for 2K2 and cycles

Given simple graphs F, G, and H, we say that F → (G, H) if, for any red-blue coloring of the edges of F, there exists either a red-colored copy of G or a blue-colored copy of H. The Ramsey number r(G, H) is the smallest integer r such that the complete graph K r satisfies K r → (G, H). The size Ramsey number rˆ(G, H) is the smallest number of edges in a graph F that satisfies F → (G, H). In addition, we distinguish two key variations of size Ramsey numbers: the restricted size Ramsey number and the connected size Ramsey number. The restricted version limits the graph F to a certain class of graphs, while the connected variant requires that the Ramsey configuration must appear in a connected subgraph of F. In this paper, we investigate algorithmic techniques for computing the restricted and connected size Ramsey numbers for 2K 2 versus cycles. We introduce a novel approach that combines a reinforcement learning-based local search (RLS) method with a verification procedure using Nauty. Our results extend previous research in Ramsey theory by providing new exact values for both restricted and connected cases.