New bounds for Ramsey numbers [formula omitted

Abstract

Let R(H1,H2) denote the Ramsey number for the graphs H1,H2, and let Jk be Kk−e. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: 49≤R(K3,J12), 36≤R(J4,K8), 43≤R(J4,J10), 52≤R(K4,J8), 37≤R(J5,J6), 43≤R(J5,K6), 65≤R(J5,J7). We also use a gluing strategy to derive a new upper bound on R(J5,J6). With both strategies combined, we prove the value of two Ramsey numbers: R(J5,J6)=37 and R(J5,J7)=65. We also show that the 64-vertex extremal Ramsey graph for R(J5,J7) is unique. Furthermore, our algorithms also allow the establishment of new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: R(W7,W4)=21, R(W7,W7)=19, R(K3,4,K3,4)=25, and R(K3,5,K3,5)=33.