Question

4. Approximating Clique. The Maximum Clique problem is to compute a clique (i.e., a complete subgraph) of maximum size in a given undirected graph G. Let G = (V,E) be an undirected graph. For any integer k ≥ 1, define G(k) to be the undirected graph (V (k), E(k)), where V (k) is the set of all ordered k-tuples of vertices from V , and E(k) is defined so that (v1,v2,...,vk) is adjacent to (w1,w2,...,wk) if and only if, for each i, 1 ≤ i ≤ k, either vertex vi is adjacent to wi in G, or vi = wi.

(a) Prove that the size of a maximum clique in G(k) is equal to the kth power of the size of a maximum clique in G.

Answer 1

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