ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


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

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.


[1]N.R. Aravind, N. Narayanan and C.R. Subramanian. Oriented colouring of some graph products, Discuss. Math. Graph Theory 31 (2011) 675--686, doi: 10.7151/dmgt.1572 .
[2]N.R. Aravind and C.R. Subramanian. Forbidden subgraph colorings and the oriented chromatic number, in: Proc. 20th Int. Workshop on Combinatorial Algorithms, IWOCA'09, Lecture Notes in Comput. Sci. 5874 (2009) 60--71, doi: 10.1007/978-3-642-10217-2_9.
[3]L. Esperet and P. Ochem, Oriented colorings of 2-outerplanar graphs, Inform. Proc. Letters 101 (2007) 215--219, doi: 10.1016/j.ipl.2006.09.007.
[4]G. Fertin, A. Raspaud and A. Roychowdhury, On the oriented chromatic number of grids, Inform. Proc. Letters 85 (2003) 261--266, doi: 10.1016/S0020-0190(02)00405-2.
[5]W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (John Wiley & Sons, New York, 2000).
[6]A.V. Kostochka, É. Sopena and X. Zhu, Acyclic and oriented chromatic numbers of graphs, J. Graph Theory 24 (1997) 331--340, doi: 10.1002/(SICI)1097-0118(199704)24:4<331::AID-JGT5>3.0.CO;2-P .
[7]J.W. Moon, Topics on Tournaments (Holt, Rinehart and Winston, New York, 1968).
[8]P. Ochem, Oriented colorings of triangle-free planar graphs, Inform. Proc. Letters 92 (2004) 71--76, doi: 10.1016/j.ipl.2004.06.012.
[9]P. Ochem. Negative results on acyclic improper colorings, Proc. Euro Comb'05, Discrete Math. Theoret. Comput. Sci., Conference Volume AE (2005) 357--362.
[10]A. Pinlou and É. Sopena, Oriented vertex and arc colorings of outerplanar graphs, Inform. Proc. Letters 100 (2006) 97--104, doi: 10.1016/j.ipl.2006.06.012.
[11]A. Raspaud and É. Sopena, Good and semi-strong colorings of oriented planar graphs, Inform. Proc. Letters 51 (1994) 171--174, doi: 10.1016/0020-0190(94)00088-3.
[12]É. Sopena, Oriented graph coloring, Discrete Math. 229 (2001) 359--369, doi: 10.1016/S0012-365X(00)00216-8.
[13]É. Sopena and L. Vignal, A note on the oriented chromatic number of graphs with maximum degree three, Research Report (1996),
[14]D.R. Wood, On the oriented chromatic number of dense graphs, Contributions to Discrete Math. 2 (2007) 145--152.

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