Chromatic number and vertex covering number For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$. Let $\chi(G)$ denote the chromatic number of $G$.
Is there a graph $G$ with $\tau(G) < \chi(G) - 1$?

Let $S$ be a minimum vertex cover. Give every vertex of $S$ a different color. Finally give all vertices of $V-S$ the same, new color.
I trust, you can take it from here.
