Abstract
Graph coloring is an assignment of labels, so-called colors, to the objects of a graph subject to certain constraints. The coloring problem on undirected graphs has been well studied, as it is one of the most fundamental problems. Meanwhile, there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G is a partition of the vertices 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. Even deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. This motivates to consider the problem on special graph classes.
In this paper we consider two different recursively defined classes of oriented graphs, namely msp-digraphs (short for minimal series-parallel digraphs) and oriented co-graphs (short for oriented complement reducible graphs).
We show that every msp-digraph has oriented chromatic number at most 7 and give an example that this is best possible. We use this bound together with the recursive structure of msp-digraphs to give a linear time solution for computing the oriented chromatic number of msp-digraphs.
Further, we use the concept of perfect orderable graphs in order to show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring, which generalizes a known solution for the oriented coloring problem on oriented co-graphs.
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Notes
- 1.
DET is the class of decision problems which are reducible in logarithmic space to the problem of computing the determinant of an integer valued \(n\times n\)-matrix.
- 2.
In [29], the tree-structure for an msp-digraphs is denoted as binary decomposition tree.
- 3.
Please note that using colors starting at value 0 instead of 1 does not contradict Definition 3.
- 4.
We implemented an algorithm which takes an oriented graph G and an integer k as an input and which decides whether \(\chi _o(G)\le k\).
- 5.
The proofs of the results marked with a \(\bigstar \) are omitted due to space restrictions.
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Acknowledgments
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 388221852.
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Gurski, F., Komander, D., Lindemann, M. (2020). Oriented Coloring of msp-Digraphs and Oriented Co-graphs (Extended Abstract). In: Wu, W., Zhang, Z. (eds) Combinatorial Optimization and Applications. COCOA 2020. Lecture Notes in Computer Science(), vol 12577. Springer, Cham. https://doi.org/10.1007/978-3-030-64843-5_50
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