Assertion: graph's orientation | /r/HasProperty | acyclic
Facets:
value: totally
type: DEGREE
value: if
type: DEGREE
Source sentences:
text: Every graph has an acyclic orientation; all acyclic orientations may be obtained by placing the vertices into a sequence, and then directing each edge from the earlier of its endpoints in the sequence to the later endpoint.
text: An orientation of an undirected graph G is totally cyclic if and only if it is a strong orientation of every connected component of G. Robbins' theorem states that a graph has a strong orientation if and only if it is 2-edge-connected; disconnected graphs may have totally cyclic orientations, but only if they have no bridges.
text: 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.
text: A graph has a k-coloring if and only if it has an acyclic orientation for which the longest path has length at most k; this is the Gallai–Hasse–Roy–Vitaver theorem (Nešetřil & Ossona de Mendez 2012).
text: Moreover, every undirected graph has an acyclic orientation, an assignment of a direction for its edges that makes it into a directed acyclic graph.
text: An orientation of an undirected graph G is totally cyclic if and only if it is a strong orientation of every connected component of G. Robbins' theorem states that a graph has a strong orientation if and only if it is 2-edge-connected; disconnected graphs may have totally cyclic orientations, but only if they have no bridges.