A Study of $k$-dipath Colourings of Oriented Graphs

TL;DR

This paper investigates $k$-dipath colourings in oriented graphs, establishing complexity dichotomies and extending homomorphism models for fixed parameters, advancing understanding of graph colouring constraints.

Contribution

It introduces a homomorphism model for $k$-dipath colourings and proves complexity dichotomies for fixed $k$ and $t$, extending prior work for $k=2$.

Findings

Dichotomy theorems for the complexity of $k$-dipath colouring decision problems.

Extension of Sherk's homomorphism model to general $k$.

Identification of cases where the problem is polynomial-time solvable or NP-complete.

Abstract

We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case $k=2$ is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed $k$ and $t$, whether there exists such a $t$-colouring are proved.