# Oriented coloring on recursively defined digraphs

## TL;DR

This paper studies oriented graph coloring, focusing on a special class called oriented co-graphs, and provides efficient algorithms for coloring, computing the chromatic number, and testing isomorphism.

## Contribution

It introduces characterizations and a linear-time algorithm for optimal coloring of oriented co-graphs, and relates the chromatic number to the longest oriented path.

## Key findings

- Linear-time algorithm for optimal oriented coloring of oriented co-graphs
- Oriented chromatic number equals longest oriented path length plus one in oriented co-graphs
- Linear-time solution for graph isomorphism problem on oriented co-graphs

## Abstract

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G=(V,A) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the chromatic number of an oriented graph is an NP-hard problem. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.