# The character graph of a finite group is perfect

## TL;DR

This paper proves that the character graph of any finite group, based on irreducible character degrees, is always perfect and that its complement has a chromatic number at most three, linking group theory and graph theory.

## Contribution

The paper establishes that the character graph of a finite group is always perfect and bounds the chromatic number of its complement, providing new insights into the structure of character graphs.

## Key findings

- The character graph of any finite group is perfect.
- The chromatic number of the complement of the character graph is at most three.
- Links between group theory and perfect graph theory are demonstrated.

## Abstract

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In graph theory, a perfect graph is a graph $\Gamma$ in which the chromatic number of every induced subgraph $\Delta$ of $\Gamma$ equals the clique number of $\Delta$. In this paper, we show that the character graph $\Delta(G)$ of a finite group $G$ is always a perfect graph. We also prove that the chromatic number of the complement of $\Delta(G)$ is at most three.