# Chi-boundedness of graph classes excluding wheel vertex-minors

## TL;DR

This paper proves that graph classes excluding wheel vertex-minors are hi-bounded, extending previous results to a broader class and establishing a key property related to graph coloring and structure.

## Contribution

It establishes hi-boundedness for graphs with no wheel vertex-minors, generalizing prior results for circle graphs and specific wheel minors.

## Key findings

- Graphs with no wheel vertex-minors are hi-bounded.
- Extends hi-boundedness to broader classes of graphs.
- Includes previous results as special cases.

## Abstract

A class of graphs is $\chi$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $\chi$-bounded. This generalizes several previous results; $\chi$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.