## 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 $$\bar G$$ 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 $$R(C_n,K_r) \leq nr^2$$ for all $$r$$ and $$n$$ and that $$(R(C_n,K_r)=(r-1)(n-1)+1$$ if $$n \geq r^2-2$$. Let $$K^{t+1}_r$$ denote the complete $$(t+1)$$-partite graph $$K(r_1, \ldots ,r_{t+1})$$ for which $$r_i=r$$ for each $$i$$. Then $$R(C_n,K^{t+1}_r)=t(n-1)+r$$ for sufficiently large $$n$$.
Reviewer: G.Chartrand

### MSC:

 05C35 Extremal problems in graph theory 05C15 Coloring of graphs and hypergraphs

### Citations:

Zbl 0294.05122; Zbl 0261.05118; Zbl 0261.05119
Full Text:

### References:

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