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| Paper IPM / M / 18319 |
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| Abstract: | |
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For given graphs $G_1$, $G_2$ and $G$, by $G\rightarrow (G_1, G_2)$ we mean if the edges of $G$ are arbitrarily colored by red and blue, then there is either a red monochromatic copy of $G_1$ or a blue monochromatic copy of $G_2$ in $G$. The Ramsey number $R(G_1, G_2)$ is defined as the smallest positive integer $n$ such that $K_n\rightarrow (G_1, G_2)$ and the star-critical Ramsey number $R_*(G_1, G_2)$ is defined as the $\min\{\delta(G):~G\subseteq K_{R(G_1, G_2)}, ~G\rightarrow (G_1, G_2)\}$. The size Ramsey number $\hat{R}(G_1, G_2)$ is defined as the minimum number of edges of a graph $G$ such that $G\rightarrow(G_1, G_2)$. In this paper, the star-critical Ramsey number of a forest versus a near complete graph will be computed exactly and a sharp bound is given for their size Ramsey numbers.
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