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Analogues of cliques for oriented coloring

100%

Klostermeyer W., MacGillivray G.

Discussiones Mathematicae Graph Theory

2004

tom 24

nr 3

373-387

We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.

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

100%

Sopena É.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 3

517-533

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.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

1 Klostermeyer W.

1 MacGillivray G.

1 Sopena É.

1 2012

1 2004

Wyniki wyszukiwania

Analogues of cliques for oriented coloring

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