# The intersection of two vertex coloring problems

## TL;DR

This paper investigates the complexity of coloring graphs that are free of certain induced subgraphs, specifically focusing on the intersection of two unresolved vertex coloring problems involving even holes and specific forbidden subgraphs.

## Contribution

The paper provides partial results on the complexity of coloring {4K1, C4, C6}-free graphs, an intersection of two open problems in graph coloring.

## Key findings

- Partial results on coloring {4K1, C4, C6}-free graphs
- Insights into the complexity of coloring even hole-free graphs
- Insights into the complexity of coloring {4K1, C4}-free graphs

## Abstract

A hole is an induced cycle with at least four vertices. A hole is even if its number of vertices is even. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. Currently, the following two problems are unresolved: the complexity of coloring even hole-free graphs, and the complexity of coloring {4K1, C4}-free graphs. The intersection of these two problems is the problem of coloring {4K1, C4, C6}-free graphs. In this paper we present partial results on this problem.