Suppose you are given a bipartite graph (L, R, E), where L denotes the vertices
ID: 3676034 • Letter: S
Question
Suppose you are given a bipartite graph (L, R, E), where L denotes the vertices on the left, R denotes the vertices on the right and E denote the set of edges. Furthermore it is given that degree of every vertex is exactly d (you may assume that d > 0). We will construct a flow network G using this bipartite graph in the following manner: G has |L| + |R| + 2 vertices. There is a vertex corresponding to every vertex in L and R. There is also a source vertex s and a sink vertex t. There are directed edges with weight 1 from s to all vertices in L and directed edges of weight 1 from all vertices in R to t. For each edge (u, v) ? E, there is a directed edge from u to v with weight 1 in G. (The figure below shows an example of a bipartite graph and the construction of the network.)
Argue that for any such bipartite graph where the degree of every vertex is equal to d, |L| is equal to |R|.
Figure 1: An example bipartite graph (with d-2) and network construction.Explanation / Answer
True
Isomorphic bipartite graphs (K1,1 K2,2 K3,3) have the same degree sequence.
For any bipartite graph where the degree of every vertex is equal call isomorphic bipartite graph.
In this |L| is equal to |R| where L denotes the vertices on the left, R denotes the vertices on the right.
Example : K3,3 has degree sequence (2,2,2) (2,2,2)
In some cases, non-isomorphic bipartite graphs may have the same degree sequence.
Example bipartite graph K3,5 with degree sequence (3,3,3) (3,3,3,3,3)
In this |L| is not equal to |R| where L denotes the vertices on the left, R denotes the vertices on the right.
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