Discussiones Mathematicae Graph Theory 32(3) (2012) 517-533
doi: 10.7151/dmgt.1624

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Upper Oriented Chromatic Number of Undirected Graphs and Oriented Colorings of Product Graphs

Éric Sopena

Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France
CNRS, LaBRI, UMR 5800, F-33400 Talence, France


The oriented chromatic number of an oriented graph G is the minimum order of an oriented graph H such that G admits a homomorphism to H. The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations.

In this paper, we introduce the new notion of the upper oriented chromatic number  of an undirected graph G, defined as the minimum order of an oriented graph U such that every  orientation G of G admits a homomorphism to U. We give some properties of this parameter, derive some general upper bounds on the ordinary and upper oriented chromatic numbers of lexicographic, strong, Cartesian and direct products of graphs, and consider the particular case of products of paths.

Keywords: product graph, oriented coloring, oriented chromatic number

2010 Mathematics Subject Classification: 05C15, 05C60.


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Received 24 September 2010
Revised 16 March 2011
Accepted 23 September 2011