  We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such
that every connected induced subgraph $H$ of $G$ uses exactly $\omega(H)$ many
colors where $\omega(H)$ is the clique number of $H$. A graph is perfectly
colorable if it admits a perfect coloring. We show that the class of perfectly
colorable graphs is exactly the class of perfect paw-free graphs. It follows
that perfectly colorable graphs can be recognized and colored in linear time.

Combinatorics