Abstract
An oriented r-coloring of an oriented graph G corresponds to an oriented graph H on r vertices, such that there exists a homomorphism from G to H. The problem of deciding whether an acyclic digraph allows an oriented 4-coloring is already NP-hard. The oriented chromatic number of an oriented graph G is the smallest integer r such that G allows an oriented r-coloring.
In this paper we consider msp-digraphs (short for minimal series-parallel digraphs), which can be defined from the single vertex graph by applying the parallel composition and series composition. In order to show several results for coloring msp-digraphs, we introduce the concept of oriented colorings excluding homomorphisms to digraphs H with short cycles. A g-oriented r-coloring of an oriented graph G is a homomorphism from G to some digraph H on r vertices of girth at least \(g+1\). The g-oriented chromatic number of G is the smallest integer r such that G allows a g-oriented r-coloring.
As our main result we show that for every msp-digraph the g-oriented chromatic number is at most \(2^{g+1}-1\). We use this bound together with the recursive structure of msp-digraphs to give a linear time solution for computing the g-oriented chromatic number of msp-digraphs. This implies that every msp-digraph has oriented chromatic number at most 7. Furthermore, we conclude that the chromatic number of the underlying undirected graphs of msp-digraphs can be bounded by 3. Both bounds are best possible and the exact chromatic numbers can be computed in linear time. Finally, we conclude that k-power digraphs of msp-digraphs have oriented chromatic number at most \(2^{2k+1}-1\).
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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.
While within oriented graph coloring only homomorphisms to oriented graphs are important, in Definition 2 we consider colorings of oriented graphs G where the color graph H is only oriented for \(g\ge 2\).
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This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 388221852.
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Gurski, F., Komander, D., Lindemann, M. (2021). Homomorphisms to Digraphs with Large Girth and Oriented Colorings of Minimal Series-Parallel Digraphs. In: Uehara, R., Hong, SH., Nandy, S.C. (eds) WALCOM: Algorithms and Computation. WALCOM 2021. Lecture Notes in Computer Science(), vol 12635. Springer, Cham. https://doi.org/10.1007/978-3-030-68211-8_15
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