Paths With Three Blocks In Digraphs

TL;DR

This paper investigates the existence of specific three-block paths in directed graphs with high chromatic number, establishing bounds and conditions for their presence.

Contribution

It introduces new bounds for the existence of paths with three blocks in digraphs based on chromatic number, including quadratic bounds for certain path structures.

Findings

Any (2k+1)-chromatic digraph contains a P(1,k,1) path.

Existence of P(1,l,1) with l ≥ k in (k+4)-chromatic digraphs.

Established quadratic bounds for paths with three blocks.

Abstract

A path with three blocks $P(k,l,r)$ is an oriented path formed by $k$-forward arcs followed by $l$-backward arcs then $r$-forward arcs. We prove that any $(2k+1)$-chromatic digraph contains a path $P(1,k,1)$. However the existence of $P(1,l,1)$ with $l \geq k$ is established in any $(k+4)$-chromatic digraph. In general, we establish a quadratic bound for paths with three blocks.