# Coloring graphs with no induced subdivision of $K_4^+$

## TL;DR

This paper proves that graphs with a large gap between chromatic number and clique number necessarily contain an induced subdivision of a specific graph called $K_4^+$, advancing understanding of graph coloring constraints.

## Contribution

It establishes a new structural property linking high chromatic number to the presence of an induced subdivision of $K_4^+$ in graphs.

## Key findings

- Graphs with high chromatic number relative to clique number contain an induced $K_4^+$ subdivision.
- Provides a structural characterization connecting coloring and induced subgraphs.
- Enhances understanding of the relationship between chromatic number and graph subdivisions.

## Abstract

Let $K_4^+$ be the 5-vertex graph obtained from $K_4$, the complete graph on four vertices, by subdividing one edge precisely once (i.e. by replacing one edge by a path on three vertices). We prove that if the chromatic number of some graph $G$ is much larger than its clique number, then $G$ contains a subdivision of $K_4^+$ as an induced subgraph.