Question

In: Advanced Math

Let G be a bipartite graph and r ∈ Z>0. Prove that if G is r-regular,...

Let G be a bipartite graph and r ∈ Z>0. Prove that if G is r-regular, then G has a perfect matching.

HINT: Use the Marriage Theorem and the Pigeonhole Principle. Recall that G is r-regular means every vertex of G has degree

Solutions

Expert Solution

To prove this result we will required two results:

Result 1:

Hall's Theorem:

Let G be a bipartite graph with bipartition . Then G contains a matching that saturates every vertex in if and only if .

Result 2:

If is a K-regular bipartite graph with K > 0 then where and are bipartition of .

Suppose is a K-regular bipartite graph with bipartition and

......................By Result 2

Let then

Suppose and are the sets of edges incident with vertices in and respectively.

Clearly by definition of ,

and Hence by Hall's theorem has a matching M saturating every vertex in .

M is perfect matching

has a perfect Matching.

Colby Messinger answered 1 year ago

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