Perfectly Colorable Graphs

TL;DR

This paper introduces the concept of perfect coloring in graphs, characterizes perfectly colorable graphs as exactly the perfect paw-free graphs, and shows they can be recognized and colored efficiently.

Contribution

It defines perfect coloring, characterizes perfectly colorable graphs as perfect paw-free graphs, and provides a linear-time recognition and coloring algorithm.

Findings

Perfect coloring requires each connected induced subgraph to use exactly its clique number of colors.

Perfectly colorable graphs are exactly the class of perfect paw-free graphs.

Recognition and coloring of these graphs can be done in linear time.

Abstract

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.