In graph theory, an acyclic orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that does not form any directed cycle and therefore makes it into a directed acyclic graph. Every graph has an acyclic orientation.
The chromatic number of any graph equals one more than the length of the longest path in an acyclic orientation chosen to minimize this path length. Acyclic orientations are also related to colorings through the chromatic polynomial, which counts both acyclic orientations and colorings.
The planar dual of an acyclic orientation is a totally cyclic orientation, and vice versa. The family of all acyclic orientations can be given the structure of a partial cube by making two orientations adjacent when they differ in the direction of a single edge.
Orientations of trees are always acyclic, and give rise to polytrees. Acyclic orientations of complete graphs are called transitive tournaments. The bipolar orientations are a special case of the acyclic orientations in which there is exactly one source and one sink; every transitive tournament is bipolar.
Every graph has an acyclic orientation. One way to generate an acyclic orientation is to place the vertices into a sequence, and then direct each edge from the earlier of its endpoints in the sequence to the later endpoint.
The vertex sequence then becomes a topological ordering of the resulting directed acyclic graph (DAG), and every topological ordering of this DAG generates the same orientation.
Because every DAG has a topological ordering, every acyclic orientation can be constructed in this way.
However, it is possible for different vertex sequences to give rise to the same acyclic orientation, when the resulting DAG has multiple topological orderings.
For instance, for a four-vertex cycle graph (shown), there are 24 different vertex sequences, but only 14 possible acyclic orientations.
The Gallai–Hasse–Roy–Vitaver theorem states that a graph has an acyclic orientation in which the longest path has at most k vertices if and only if it can be colored with at most k colors.
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vignette|Par suppression d'une arête rouge arbitraire, ce cycle hamiltonien donne une chaîne de longueur maximale. En théorie des graphes et en informatique théorique, le problème de la plus longue chaîne (ou le problème du plus long chemin dans le cas d'un graphe orienté) consiste à déterminer la plus longue chaîne élémentaire dans un graphe. Une chaîne est élémentaire si elle ne passe pas deux fois par le même sommet. La longueur d'une chaîne peut être mesurée par le nombre d'arêtes qui la composent ou, dans le cas de graphes pondérés, par la somme des poids des arêtes du chemin.
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. A directed graph is called an oriented graph if none of its pairs of vertices is linked by two symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of (x, y) and (y, x) may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree.
Le polynôme de Tutte, aussi appelé polynôme dichromatique ou polynôme de Tutte–Whitney, est un polynôme invariant de graphes dont les valeurs expriment des propriétés d'un graphe. C'est un polynôme en deux variables qui joue un rôle important en théorie des graphes et en combinatoire. Il est défini pour tout graphe non orienté et contient des informations liées à ses propriétés de connexité. L'importance de ce polynôme provient des informations qu'il contient sur le graphe .
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