Melnikov's Valency Variety Problem

Source abstract

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 graph $G$ with at least two vertices satisfies $$\begin{equation*} χ(G)\ge\left\lceil1+\frac{2w(G)(w(G)-1)}{4w(G)(|V(G)|-w(G))+(|V(G)|-w(G)-1)^2}\right\rceil, \end{equation*}$$ and consequently $$\begin{equation*} χ(G)\ge\left\lceil\frac{\lfloor w(G)/2\rfloor}{|V(G)|-w(G)}\right\rceil. \end{equation*}$$ Finally, we also construct an explicit infinite family of graphs attaining equality in both bounds. In particular, these examples show that Melnikov's proposed strict inequality is false and that the bounds above are best possible.