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≥k is established in any (k+4)-chromatic digraph. In general, we establish a quadratic bound for paths with three blocks.
@article{arxiv.2110.09933,
title = {Paths With Three Blocks In Digraphs},
author = {Maidoun Mortada and Amine El Sahili and Zahraa Mohsen},
journal= {arXiv preprint arXiv:2110.09933},
year = {2021}
}