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Claim. Let G = (W, E) be a (simple, undirected) graph with |W| 2 5. Then there are two distinct subsets of nodes U, V C W with |U| = [V| = 2 such that ExEv deg(x) = >rev deg(x). (I.e. the sum of the degrees of the two nodes in U is the same as that of the two nodes in V.) Use the Pigeonhole Principle to fill out the following proof of the claim. Note that . There are other ways of proving this claim, but this problem asks you to use the PHP. . You may use the fact that for all n > 5, "- > 2n -1, if it helps. . Any variable that represents a set should have an uppercase variable name. Proof. We will prove this claim using the Pigeonhole Principle. . Let the pigeons (A) be the set . Let the pigeonholes (B) be the set . Let f : A -> B be defined by f( ) := Note that f is a well-defined function: (1) for any pigeon _ E f ( ) = is computable because (2) for any pigeon _ E if = b and = c then b = c, because (3) for any pigeon _ E f ( ) = is within the codomain because Mathematical Statement Reason this Statement is True (From the Approved List) |A| = because A = |B= because B = 3 _, _ E A : [ ( # _ ) ( 0) = f() )] (i.e. pigeons _ _ with same pigeonhole) by the def of A and f
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