Ramsey Good Graphs with Long Suspended Paths

Abstract

A connected graph H with $|H|\ge \sigma (G)$ is said to be G-good if $R(G,H)=(\chi (G)-1)(|H|-1)+\sigma (G)$. For an integer $\ell \ge 3$, let $P_\ell$ be a path of order $\ell$, and $H^{(\ell )}$ a graph obtained from H by joining the end vertices of $P_\ell$ to distinct vertices u, v of H. It is widely known that for any graphs G and H, if $\ell$ is sufficiently large, then $H^{(\ell )}$ is G-good. In this note, we show that there exists a constant $c=c(\Delta )$ such that for any graphs G and H with $\Delta (G)\le \Delta$ and $\Delta (H)\le \Delta$, if $\ell \ge c\cdot (|G|+|H|)$, then $H^{(\ell )}$ is G-good; and if $n\ge 2\alpha (G)+\Delta ^2(G)+4$, then $P_n$ is G-good.