For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a [vertex cover][1] of $G$. 

As noted [here][2], we have $\tau(G) \geq \chi(G) - 1$ for all finite graphs $G$. I'm interested in graphs $G$ with $\tau(G) = \chi(G) - 1$. I found only examples of such graphs where $\chi(G) = \omega(G)$ (where $\omega(G)$ is the clique number of $G$).

**Question**: Is there a graph $G$ with $\omega(G) < \chi(G)$ and $\tau(G) = \chi(G) - 1$?


  [1]: http://en.wikipedia.org/wiki/Vertex_cover
  [2]: http://math.stackexchange.com/questions/1123167/chromatic-number-and-vertex-covering-number
  [4]: http://en.wikipedia.org/wiki/Clique_%28graph_theory%29