# Triangle-free graphs that do not contain an induced subdivision of $K_4$ are 3-colorable

## TL;DR

This paper proves that triangle-free graphs lacking an induced subdivision of K4 are always 3-colorable, confirming a conjecture and advancing understanding of graph coloring constraints.

## Contribution

It establishes that such graphs are 3-colorable, resolving a conjecture by Trotignon and Vuskovic.

## Key findings

- Triangle-free graphs without induced K4 subdivisions are 3-colorable.
- Confirmed the conjecture of Trotignon and Vuskovic.
- Provides a structural characterization relevant to graph coloring.

## Abstract

We show that triangle-free graphs that do not contain an induced subgraph isomorphic to a subdivision of K4 are 3-colorable. This proves a conjecture of Trotignon and Vuskovic.