On triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph

TL;DR

This paper proves a decomposition theorem for a specific class of triangle-free graphs that lack a certain subdivision, and shows that graphs with girth at least 5 in this class are 3-colorable.

Contribution

It introduces a new decomposition theorem for triangle-free graphs without a K4 subdivision as an induced subgraph, and establishes 3-colorability for graphs with girth at least 5 in this class.

Findings

Decomposition theorem for the class of triangle-free graphs without a K4 subdivision as an induced subgraph.

Graphs with girth at least 5 in this class are 3-colorable.

Provides structural insights into these graphs.

Abstract

We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least~5 in this class is 3-colorable.