On the Ramsey-Goodness of Paths.

For a graph G, we denote by $$\nu (G)$$ the order of G, by $$\chi (G)$$ the chromatic number of G and by $$\sigma (G)$$ the minimum size of a color class over all proper $$\chi (G)$$ -colorings of G. For two graphs $$G_1$$ and $$G_2$$ , the Ramsey number $$R(G_1,G_2)$$ is the least integer r such that for every graph G on r vertices, either G contains a $$G_1$$ or $$\overline{G}$$ contains a $$G_2$$ . Suppose that $$G_1$$ is connected. We say that $$G_1$$ is $$G_2$$ -good if $$R(G_1,G_2)=(\chi (G_2)-1)(\nu (G_1)-1)+\sigma (G_2)$$ . In this note, we obtain a condition for graphs H such that a path is H-good. [ABSTRACT FROM AUTHOR]