Is a graph G bipartite?
Theorem 1 A graph G is bipartite if and only if it does not contain any cycle of odd length. Proof: ()) Easy: each cycle alternates between left-to-right edges and right-to-left edges, so it must have an even length.
How to generalize the previous theorem?
We can generalize the previous theorem by saying that everyk-partite graph isk-colorable and the proof is similar to the proof for two. Similar to the idea of coloring, we have that amatching MinGis a set of edges such that no two edges share a common vertex. Another way to say this is that the set of edges must be pairwise non-adjacent.
Is GIS bipartite?
If Gis bipartite, then it is possible to assign colors red and blue to the vertices ofGin such a way, that no two vertices of the same color are adjacent. (v). Gis bipartite if and only if each of its components is bipartite. Theorem. A graph Gis bipartite if and only if it has no odd cycles. Proof. First, suppose that Gis bipartite.
What is the perfect matching theorem?
Theorem 2.10 (Perfect Matching)A perfect matching exists on a bipartite graph G with bipartition X and Y if and only if for every subset S of X we have |S| ? |N(S)| and |X|=|Y|. That is, if for every subset S of X, the number of elements in S is less than or equal to the number of elements in the neighborhood of S. Proof.