Papers › New bounds for Ramsey numbers R(Kₖ-e,Kₗ-e)
New bounds for Ramsey numbers R(Kₖ-e,Kₗ-e)
Jan Goedgebeur, Steven Van Overberghe
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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 to establish 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(K_(3,4),K_(3,4)) = 25, and R(K_(3,5), K_(3,5))=33.
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