For graphs G and H, the Ramsey number R(G,H) is the least integer n such that every 2-coloring of the edges of K_n contains a subgraph isomorphic to G in the first color or a subgraph isomorphic to H in the second color. Graph G is a (C_4, K_n)-graph if G doesn't contain a cycle C_4 and G has no independent set of order n. Jayawardene and Rousseau showed that 21 <= R(C_4,K_7) <= 22. In this work we determine R(C_4, K_7) = 22 and R(C_4,K_8) = 26, and enumerate various families of (C_4, K_n)-graphs. In particular, we construct all (C_4, K_n)-graphs for n < 7, and all (C_4,K_7)-graphs on at least 19 vertices. Most of the results are based on computer algorithms.
Department, Program, or Center
Center for Advancing the Study of CyberInfrastructure
The Journal of Combinatorial Mathematics 42 (2002) 195-207
RIT – Main Campus