Colourings, Homomorphisms, and Partitions of Transitive Digraphs

TL;DR

This paper explores the computational complexity of various coloring and homomorphism problems in transitive digraphs, revealing many are intractable despite their structured nature, and highlights open problems in the field.

Contribution

It provides new insights into the complexity classification of coloring and homomorphism problems in transitive digraphs, including some motivational results and open questions.

Findings

Many problems are computationally intractable in transitive digraphs

Structured nature of transitive digraphs does not guarantee tractability

The paper identifies open problems in the complexity classification

Abstract

We investigate the complexity of generalizations of colourings (acyclic colourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), for the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.