Computation of the Ramsey Number $R(W_5,K_5)$

We determine the value of the Ramsey number $R(W_5,K_5)$ to be 27, where $W_5 = K_1 + C_4$ is the 4-spoked wheel of order 5. This solves one of the four remaining open cases in the tables given in 1989 by George R. T. Hendry, which included the Ramsey numbers $R(G,H)$ for all pairs of graphs $G$ and $H$ having five vertices, except seven entries. In addition, we show that there exists a unique up to isomorphism critical Ramsey graph for $W_5$ versus $K_5$. Our results are based on computer algorithms.

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