For n 1, define the graph H, with vertex set V(H) = {1,2,3,..., n} and suppose...

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For n 1, define the graph H, with vertex set V(H) = {1,2,3,..., n} and suppose xy = E(H) if and only if x + y is a prime number. (a) Prove that H is bipartite for n 1. (b) Find the value of n so that H is planar but H+1 is non-planar. For your value of n, you must give a planar drawing of H and also a proof that H+1 is non-planar. (c) Determine x(Hn) for each n 1. For n 1, define the graph H, with vertex set V(H) = {1,2,3,..., n} and suppose xy = E(H) if and only if x + y is a prime number. (a) Prove that H is bipartite for n 1. (b) Find the value of n so that H is planar but H+1 is non-planar. For your value of n, you must give a planar drawing of H and also a proof that H+1 is non-planar. (c) Determine x(Hn) for each n 1.