Let GG be a graph on nn vertices. We call a subset AA of the vertex set V(G)V(G)kk-small if, for every vertex v∈Av∈A, deg(v)≤n−|A|+kdeg(v)≤n−|A|+k. A subset B⊆V(G)B⊆V(G) is called kk-large if, for every vertex u∈Bu∈B, deg(u)≥|B|−k−1deg(u)≥|B|−k−1. Moreover, we denote by φk(G)φk(G) the minimum integer tt such that there is a partition of V(G)V(G) into View the MathML sourcetk-small sets, and by Ωk(G)Ωk(G) the minimum integer tt such that there is a partition of V(G)V(G) into View the MathML sourcetk-large sets. In this paper, we will show tight connections between kk-small sets, respectively kk-large sets, and the kk-independence number, the clique number and the chromatic number of a graph. We shall develop greedy algorithms to compute in linear time both φk(G)φk(G) and Ωk(G)Ωk(G) and prove various sharp inequalities concerning these parameters, which we will use to obtain refinements of the Caro–Wei Theorem, Turán’s Theorem and the Hansen–Zheng Theorem among other things.

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