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Melnikov's Valency Variety Problem
Let $w(G)$ denote the number of distinct vertex degrees of a finite simple graph $G$. Melnikov asked for a lower bound on the chromatic number $χ(G)$ in terms of $|V(G)|$ and $w(G)$, and conjectured a strict bound of this type. We resolve Melnikov's valency-variety problem by proving that every grap
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