Computation of Some Generalized Ramsey Numbers

For given graphs \( G_1 \) and \( G_2 \), the \( 2 \)-color Ramsey number \( R(G_1, G_2) \) is defined to be the least positive integer \( n \) such that every \( 2 \)-coloring of the edges of the complete graph \( K_n \) contains a copy of \( G_1 \) colored with the first color or a copy of \( G_2 \) colored with the second color. In this note, we obtained some new exact values of generalized Ramsey numbers such as cycle versus book, book versus book, and complete bipartite graph versus complete bipartite graph.