Partitions of graphs into small and large sets

Let $G$ be a graph on $n$ vertices. We call a subset $A$ of the vertex set $V(G)$ \emph{$k$-small} if, for every vertex $v\in A$, $\deg(v) \le n - |A| + k$\mathrm{.}Asubset$B \subseteq V(G)is called \emph{k-large} if, for every vertexu \in B$,$\deg(u) \ge |B| - k - 1$\mathrm{.}Moreover,wedenoteby$\varphi_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-small sets, and by ${\mathrm{\Omega }}_k(G)$ the minimum integer $t$ such that there is a partition of $V(G)$ into $t$ $k$-large sets. In this paper, we will show tight connections between $k$-small sets, respectively $k$-large sets, and the $k$-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both ${\phi }_k(G)$ and ${\mathrm{\Omega }}_k(G)$ and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro-Wei Theorem, the Tur'an Theorem and the Hansen-Zheng Theorem among other things.