$\delta_k$-small sets in graphs

Abstract

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 $\alpha(G)$.