# Induced subgraphs of graphs with large chromatic number. VI. Banana trees

## TL;DR

This paper characterizes certain complex graphs called bananas, showing they appear as induced subdivisions in large-chromatic graphs with bounded clique number, extending previous results on trees and cycles.

## Contribution

It generalizes earlier work by proving that graphs formed from trees by replacing edges with bananas also have the induced subdivision property.

## Key findings

- Bananas are induced subgraphs in graphs with large chromatic number.
- Graphs formed from trees by replacing edges with bananas have the property.
- Other multigraphs with similar properties are identified.

## Abstract

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that every tree has this property; and in another earlier paper with M. Chudnovsky, we proved that every cycle has this property. Here we give a common generalization. Say a banana is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentioned. We also find some other multigraphs with the same property.