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Ramsey numbers for cycles in graphs. (English) Zbl 0248.05127
For two graphs G 1 and G 2 , the Ramsey number R(G 1 ,G 2 ) is the minimum p such that for any graph G of order p, either G 1 is a subgraph of G of G 2 is a subgraph of the complement G ¯ of G. The authors determine the Ramsey numbers in the cases where G 1 and G 2 are certain cycles. [These Ramsey numbers have since been established completely by J. Faudree and R. H. Schelp [Discrete Math. 8, 313-329 (1974; Zbl 0294.05122)] and V. Rosta [J. Comb. Theory, Ser. B 15, 94-104, 105-120 (1973; Zbl 0261.05118 and Zbl 0261.05119)]. The authors show that R(C n ,K r )nr 2 for all r and n and that (R(C n ,K r )=(r-1)(n-1)+1 if nr 2 -2. Let K r t+1 denote the complete (t+1)-partite graph K(r 1 ,...,r t+1 ) for which r i =r for each i. Then R(C n ,K r t+1 )=t(n-1)+r for sufficiently large n.
Reviewer: G.Chartrand

MSC:
05C35Extremal problems (graph theory)
05C15Coloring of graphs and hypergraphs