An Inequality for Generalized Chromatic Graphs

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