# In absence of long chordless cycles, large tree-width becomes a local phenomenon

## TL;DR

This paper proves that graphs with sufficiently large tree-width necessarily contain either large bipartite subgraphs or long chordless cycles, revealing a local-global structural property.

## Contribution

It establishes a new relationship between large tree-width and the presence of specific subgraphs, specifically long chordless cycles or complete bipartite graphs.

## Key findings

- Graphs with large tree-width contain long chordless cycles or large bipartite subgraphs.
- The result links global tree-width to local subgraph structures.
- Provides a structural characterization relevant for graph theory and algorithms.

## Abstract

We prove that, for all $\ell$ and $s$, every graph of sufficiently large tree-width contains either a complete bipartite graph $K_{s,s}$ or a chordless cycle of length greater than $\ell$.