New bounds for Ramsey numbers R(K_k-e,K_l-e)
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Let R(H_1,H_2) denote the Ramsey number for the graphs H_1, H_2, and let J_k be K_k-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(K_3,J_12), 36 ≤ R(J_4,K_8), 43 ≤ R(J_4,J_10), 52 ≤ R(K_4,J_8), 37 ≤ R(J_5,J_6), 43 ≤ R(J_5,K_6), 65≤ R(J_5,J_7). We also use a gluing strategy to derive a new upper bound on R(J_5,J_6). With both strategies combined, we prove the value of two Ramsey numbers: R(J_5,J_6)=37 and R(J_5,J_7)=65. We also show that the 64-vertex extremal Ramsey graph for R(J_5,J_7) is unique. Furthermore, our algorithms also allow to establish new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: R(W_7,W_4) = 21, R(W_7,W_7) = 19, R(K_3,4,K_3,4) = 25, and R(K_3,5, K_3,5)=33.
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