Assertion: graph | /r/HasProperty | bipartite
Facets:
Source sentences:
text: A graph is bipartite if and only if it contains no cycles of odd length.
text: A graph is bipartite if and only if it does not contain an odd cycle.
text: A graph is bipartite if and only if it is 2-colorable, (
text: According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph.
text: It is possible to test whether a graph is bipartite, and to return either a two-coloring (if it is bipartite) or an odd cycle (if it is not) in linear time, using depth-first search.
text: However, if the algorithm terminates without detecting an odd cycle of this type, then every edge must be properly colored, and the algorithm returns the coloring together with the result that the graph is bipartite.
text: If the algorithm terminates without finding an odd cycle in this way, then it must have found a proper coloring, and can safely conclude that the graph is bipartite.
text: For the intersection graphs of n {\displaystyle n} line segments or other simple shapes in the Euclidean plane, it is possible to test whether the graph is bipartite and return either a two-coloring or an odd cycle in time O ( n log ⁡ n )
text: The name odd cycle transversal comes from the fact that a graph is bipartite if and only if it has no odd cycles.
text: Additionally, a graph is bipartite if and only if it contains no odd cycles.