  Let $G$ be a simple $n$-vertex graph and $W\subseteq\V(G)$. We say that $W$
is a $\delta_k$-small set if $$ \sqrt[k]{\frac{\sum_{v\in W}d^k(v)}{\abs
W}}\leq n-\abs W. $$ Let $\varphi^{(k)}(G)$ denote the smallest natural number
$r$ such that $\V(G)$ decomposes into $r$ $\delta_k$-small sets, and let
$\alpha^{(k)}(G)$ denote the maximal number of vertices in a $\delta_k$-small
set of $G$. In this paper we obtain bounds for $\alpha^{(k)}(G)$ and
$\varphi^{(k)}(G)$. Since $\varphi^{(k)}(G)\leq\omega(G)\leq\chi(G)$ and
$\alpha(G)\leq\alpha^{(k)}(G)$, we obtain also bounds for the clique number
$\omega(G)$, the chromatic number $\chi(G)$ and the independence number
$\alpha(G)$.
