LOCAL AND MEAN k-RAMSEY NUMBERS FOR THE FAMILY OF GRAPHS

  • Su, Zhanjun (College of Mathematics and Information Science, Hebei Normal University) ;
  • Chen, Hongjing (Hebei Institute of Applied Mathematics) ;
  • Ding, Ren (College of Mathematics and Information Science, Hebei Normal University)
  • Published : 2009.05.31

Abstract

For a family of graphs $\mathcal{H}$ and an integer k, we denote by $R^k(\mathcal{H})$ the corresponding k-Ramsey number, which is defined to be the smallest integer n such that every k-coloring of the edges of $K_n$ contains a monochromatic copy of a graph in $\mathcal{H}$. The local k-Ramsey number $R^k_{loc}(\mathcal{H})$ and the mean k-Ramsey number $R^k_{mean}(\mathcal{H})$ are defined analogously. Let $\mathcal{G}$ be the family of non-bipartite graphs and $T_n$ be the family of all trees on n vertices. In this paper we prove that $R^k_{loc}(\mathcal{G})=R^k_{mean}(\mathcal{G})$, and $R^2(T_n)$ < $R^2_{loc}(T_n)4 = $R^2_{mean}(T_n)$ for all $n\;{\ge}\;3$.

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References

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