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For a uniformly chosen vertex $Z$, minimize the information entropy of its color over proper colorings. Multiply by the number of vertices to define this weight. Minimizing the color entropy gives the strongest contribution from that graph in the entropy weight of a graph cover inequality.

Use the same random variables for the edge cover of $G$. For a fixed color vector, any two compatible candidate vertices cannot be adjacent: an edge in a covering graph would force different colors at that coordinate. Thus the compatible candidates form an independent set and there are at most $\alpha$ of them. By the maximum entropy on a finite alphabet, $H(X\mid Y_1,\dots ,Y_{\ell })\le {\log }_2\alpha$.

Consequently the mutual information is at least ${\log }_{2}n-{\log }_2\alpha$, while the marginal and conditional-entropy bounds from part (i) make it at most $n^{-1}{\sum }_{i}w(G_i)$. The entropy weight of a graph cover bound follows: