Taking ? = 1/k in Theorem 1.1 and considering the majority color in a k-coloring of the edges of a complete graph shows that for bipartite graphs H1
Aug 8 2007 Therefore
Density theorem for bipartite graphs. Theorem: (F.-Sudakov). Let G be a bipartite graph with n vertices and maximum degree d and let H be a bipartite graph
Theorem 2. Let H be a bipartite graph with vertex classes U1 and U2.
Density theorem for bipartite graphs. Theorem: (Fox-S.) Let G be a bipartite graph with n vertices and maximum degree d and let H be a bipartite graph with
edges each pair of vertices being joined by one edge. The complete bipartite graph Km
DMTCS vol. 12:5 2010
domains is bounded by the size of a maximal bipartite minor. Key words. Graph Laplacian Nodal Domain Theorem
For any bipartite graph G a Konig covering of G is an ordered theorem for countable bipartite graphs (i.e.
Theorem 4 For a simple connected graphG the following conditions are equivalent Gis bipartite Every cycle ofG(if some) has even length F GOTTI Proof (a))(b): Assume thatGis bipartite on the partsXandY Suppose by wayof contradiction thatGhas a cycle of odd length namelyC:=v1v2: : : v2n+1v1
An Introduction to Bipartite Graphs If P is a path from the vertex v to the vertex u we refer to P as a v-u path (or often just a vu-path) If P is a v-u path say v=v 0 v 1 v 2 v k v m=u then we refer to v i v i+1 v j (for any 0!i
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.
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.
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.
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.