Oriented paths in n-chromatic digraphs

TL;DR

This thesis investigates the existence of specific oriented paths in n-chromatic digraphs, providing new proofs, correcting previous errors, and extending results for paths with two blocks.

Contribution

It offers a shorter proof of El-Sahili's theorem, clarifies approaches for paths with two blocks, and corrects a mistake in prior work.

Findings

Elementary proof of El-Sahili's theorem for antidirected paths in 5-chromatic digraphs

Analysis of approaches for paths with two blocks in n-chromatic digraphs

Correction of a mistake in Addario-Berry et al.'s proof

Abstract

In this thesis, we try to treat the problem of oriented paths in n-chromatic digraphs. We first treat the case of antidirected paths in 5-chromatic digraphs, where we explain El-Sahili's theorem and provide an elementary and shorter proof of it. We then treat the case of paths with two blocks in n-chromatic digraphs with n greater than 4, where we explain the two different approaches of Addario-Berry et al. and of El-Sahili. We indicate a mistake in Addario-Berry et al.'s proof and provide a correction for it.