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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.Peer ReviewedPostprint (published version

Let G be a simple n-vertex graph with degree sequence d1, d2, . . . , dn and vertex set V(G). The degree of v ∈ V(G) is denoted by d(v). The smallest integer r for which V(G) has an r-partitionIn this note we prove the inequality

Constacyclic codes with one and the same generator polynomial and distinct length are considered. We give a generalization of the previous result of the first author [4] for constacyclic codes. Suitable maps between vector spaces determined by the lengths of the codes are applied. It is proven that the weight distributions of the coset leaders don’t depend on the word length, but on generator polynomials only. In particular, we prove that every constacyclic code has the same weight distribution of the coset leaders as a suitable cyclic code.