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Oriented Coloring on Recursively Defined Digraphs

Algorithmics for Hard Problems Group, Institute of Computer Science, Heinrich Heine University Düsseldorf, 40225 Düsseldorf, Germany
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Algorithms 2019, 12(4), 87; https://doi.org/10.3390/a12040087
Received: 21 February 2019 / Revised: 15 April 2019 / Accepted: 22 April 2019 / Published: 25 April 2019
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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, too. 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. View Full-Text
Keywords: oriented graphs; oriented co-graphs; oriented coloring; graph isomorphism oriented graphs; oriented co-graphs; oriented coloring; graph isomorphism
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Gurski, F.; Komander, D.; Rehs, C. Oriented Coloring on Recursively Defined Digraphs. Algorithms 2019, 12, 87.

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