Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs

Éric Sopena

Discussiones Mathematicae Graph Theory, Tome 32 (2012), p. 517-533 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2012-01-01

Zbl 1257.05047

EUDML-ID : urn:eudml:doc:270820

Bibliographie

[000] [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. | Zbl 1255.05073

[001] [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. | Zbl 1267.05111

[002] [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. | Zbl 1185.05059

[003] [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. | Zbl 1173.68604

[004] [5] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (John Wiley & Sons, New York, 2000).

[005] [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 | Zbl 0873.05044

[006] [7] J.W. Moon, Topics on Tournaments (Holt, Rinehart and Winston, New York, 1968). | Zbl 0191.22701

[007] [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. | Zbl 1173.68593

[008] [9] P. Ochem. Negative results on acyclic improper colorings, Proc. Euro Comb'05, Discrete Math. Theoret. Comput. Sci., Conference Volume AE (2005) 357-362. | Zbl 1192.05056

[009] [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. | Zbl 1185.05064

[010] [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. | Zbl 0806.05031

[011] [12] É. Sopena, Oriented graph coloring, Discrete Math. 229 (2001) 359-369, doi: 10.1016/S0012-365X(00)00216-8. | Zbl 0971.05039

[012] [13] É. Sopena and L. Vignal, A note on the oriented chromatic number of graphs with maximum degree three, Research Report (1996), http://www.labri.fr/perso/sopena/.

[013] [14] D.R. Wood, On the oriented chromatic number of dense graphs, Contributions to Discrete Math. 2 (2007) 145-152. | Zbl 1203.05061