# Coloring ($P_6$, diamond, $K_4$)-free graphs

## TL;DR

This paper proves that all ($P_6$, diamond, $K_4$)-free graphs can be colored with six colors and provides an example of such a graph requiring exactly six colors, extending previous results.

## Contribution

It establishes the exact chromatic number for ($P_6$, diamond, $K_4$)-free graphs and presents a minimal example demonstrating this bound.

## Key findings

- Every ($P_6$, diamond, $K_4$)-free graph is 6-colorable.
- Existence of a ($P_6$, diamond, $K_4$)-free graph with chromatic number 6.

## Abstract

We show that every ($P_6$, diamond, $K_4$)-free graph is $6$-colorable. Moreover, we give an example of a ($P_6$, diamond, $K_4$)-free graph $G$ with $\chi(G) = 6$. This generalizes some known results in the literature.