Show that the vector is in the perfect matching polytope, Mathematics

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1.  Let G = (V,E) be a graph for which all nodes have degree 5 and where G is 5-edge is connected.

a) Show that the vector x which is indexed by the edges E and for which xe = 1/5 for all e in E is in the Perfect Matching Polytope PPM.

b)  Use your result in a) to show that G must have a perfect matching.

c) Show that b) may not be true if G is only 1-edge connected (but still has degree 5 everywhere) by giving an example of such a graph G which has no perfect matching.


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