发表机构
Fuzhou University（福州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了Melnikov关于图色数下界的猜想，证明了两个最优下界，并构造达到等号的无限图族，表明原严格不等式不成立。
AI 中文摘要
设$w(G)$表示有限简单图$G$的不同顶点度的数目。Melnikov询问了关于用$|V(G)|$和$w(G)$来给出色数$\u03c7(G)$的下界的问题，并猜想了一个这种类型的严格界。我们通过证明每个至少有两个顶点的图$G$满足\begin{equation*} \u03c7(G)\u2265\u2308\u4e00+\frac{2w(G)(w(G)-1)}{4w(G)(|V(G)|-w(G))+(|V(G)|-w(G)-1)^2}\u2309, \nonumber \u5e76\u4e14\u56e0\u6b64\u6ee1\u8db3\begin{equation*} \u03c7(G)\u2265\u2308\frac{\u230aw(G)/2\u230b}{|V(G)|-w(G)}\u2309, \nonumber \u6765\u89e3\u51b3Melnikov\u7684\u5ea6\u591a\u6837\u6027\u95ee\u9898\u3002\u6700\u540e\uff0c\u6211\u4eec\u8fd8\u6784\u9020\u4e86\u4e00\u4e2a\u660e\u786e\u7684\u65e0\u9650\u56fe\u5bb6\u65cf\uff0c\u4f7f\u5f97\u4e24\u4e2a\u4e0a\u8ff0\u754c\u90fd\u8fbe\u5230\u7b49\u53f7\u3002\u7279\u522b\u5730\uff0c\u8fd9\u4e9b\u4f8b\u5b50\u8868\u660eMelnikov\u63d0\u51fa\u7684\u4e25\u683c\u4e0d\u7b49\u5f0f\u662f\u9519\u8bef\u7684\uff0c\u4e14\u4e0a\u8ff0\u754c\u662f\u6700\u4f18\u7684\u3002
英文摘要
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.
