An Inequality for Generalized Chromatic Graphs

Let $G$ be a simple $n$-vertex graph with degree sequence $d_1,d_2,\mathrm{.}\mathrm{.}\mathrm{.},d_n$ and vertex set $\text{V}(G)$. The degree of $v\in \text{V}(G)$ is denoted by $\text{D}(v)$. The smallest integer $r$ for which $\text{V}(G)$ has an $r$-partition $$\text{V}(G)=V_1\cup V_2\cup \mathrm{.}\mathrm{.}\mathrm{.}\cup V_r,V_i\cap V_j=\mathrm{\varnothing },,i\mathrm{\ne }j$$ such that $\text{D}(v)\le n-\text{abs}{V}_i$, $\mathrm{\forall }v\in V_i$, $i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},r$ is denoted by $\text{f}(G)$. In this note we prove the inequality $$\text{f}(G)\ge \frac{n}{n-\overline{\stackrel{ˉ}{d}}},$$ where $\overline{\stackrel{ˉ}{d}}=\sqrt{{\displaystyle \frac{d_1^2+d_2^2+\mathrm{.}\mathrm{.}\mathrm{.}+d_n^2}{n}}}$.