# Bispindles in strongly connected digraphs with large chromatic number

## TL;DR

This paper explores the existence of certain bispindle subgraphs in strongly connected digraphs with large chromatic number, extending previous results and identifying new conditions under which these structures must appear.

## Contribution

It generalizes known results by constructing digraphs without specific bispindles and proves the presence of a (2,1)-bispindle in highly chromatic strongly connected digraphs.

## Key findings

- Constructed strongly connected digraphs with large chromatic number lacking (3,0) and (2,2)-bispindles.
- Proved that large chromatic number implies the existence of a (2,1)-bispindle with long paths.
- Extended the understanding of subgraph structures in high chromatic number digraphs.

## Abstract

A $(k_1+k_2)$-bispindle is the union of $k_1$ $(x,y)$-dipaths and $k_2$ $(y,x)$-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every $(1,1)$- bispindle $B$, there exists an integer $k$ such that every strongly connected digraph with chromatic number greater than $k$ contains a subdivision of $B$. We investigate generalisations of this result by first showing constructions of strongly connected digraphs with large chromatic number without any $(3,0)$-bispindle or $(2,2)$-bispindle. Then we show that strongly connected digraphs with large chromatic number contains a $(2,1)$-bispindle, where at least one of the $(x,y)$-dipaths and the $(y,x)$-dipath are long.