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The chromatic number of oriented graphs

✍ Scribed by Sopena, Eric

Publisher: John Wiley and Sons
Year: 1997
Tongue: English
Weight: 198 KB
Volume: 25
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199707)25:3<191::aid-jgt3>3.0.co;2-g

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✦ Synopsis

We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) defined as the minimum order of an oriented graph H such that G admits a homomorphism to H. We study the chromatic number of oriented k-trees and of oriented graphs with bounded degree. We show that there exist oriented k-trees with chromatic number at least 2 k+1 -1 and that every oriented k-tree has chromatic number at most (k + 1) × 2 k . For 2-trees and 3-trees we decrease these upper bounds respectively to 7 and 16 and show that these new bounds are tight. As a particular case, we obtain that oriented outerplanar graphs have chromatic number at most 7 and that this bound is tight too. We then show that every oriented graph with maximum degree k has chromatic number at most (2k -1) × 2 2k-2 . For oriented graphs with maximum degree 2 we decrease this bound to 5 and show that this new bound is tight. For oriented graphs with maximum degree 3 we decrease this bound to 16 and conjecture that there exists no such connected graph with chromatic number greater than 7.

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