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The major points which I found very interactive in learning graph theory are: Hall'S Theorem, Satisfied, Matching, Unmatched Vertex, Sequence, Distinct, Vertices, Matching Edge, Vertex, Unmatched Vertex
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Hall’s Theorem: Second Proof.
Suppose Hall’s condition is satisfied, and there is matching of A. Then there is an unmatched vertex in A. Call it a 0. We will generate a sequence of distinct vertices, a 0 , b 1 , a 1 , b 2 , a 2 ,...
In this sequence bi , ai will always be a matching edge.
bi+1 will be a neighbor of some vertex in {a 0 , a 1 ,... , ai }
Start with a 0. We can always find b 1. Can b 1 be an unmatched vertex?
Using Hall’s Theorem:
If G is k-regular (k ≥ 1 ) bipartite graph, then it has a perfect matching
To show this we just need to show that the Hall’s condition is true for k-regular bipartite graphs.
Before getting to the proof the above statement, we need to discuss some concept. First, given an undirected graph here is a way of associating a bipartite graph to it. Bipartite Double Cover of G : The two sides A and B are copies of V (G ). Lets us say the copy of the vertex i of G , is named ai in A and bi in B. We add an edge from ai to bj whenever there is an edge (i, j) in G.
In the above, note that if (i, j) is an edge in G , we get two different edges (ai , bj ) and (aj , bi ). For a directed graph, if we do a corresponding construction, a directed edge will correspond to exactly one edge in the bipartite cover graph.
To show that an undirected 2 k-regular graph G has a cycle cover ( 2 -factor) we covert G to a bipartite graph as follows. First we give a direction to each edge of G to get a directed graph G ′. Now get the bipartite cover BG ′ of G ′.
Instead of showing a cycle cover in G , we will show a directed cycle cover in G ′.
To show a directed cycle cover in G ′^ it is enough to show a perfect matching in BG ′^.
For that it is enough to show that BG ′^ is regular.
What is an Euler Tour?
How does help us to orient the edges of G ?
Finally, is it guaranteed that there is an Euler tour in G?
A connected graph G has an Euler tour if and only if every vertex has even degree.
