Independence in connected graphs

We prove that if G=(VG,EG)G=(VG,EG) is a finite, simple, and undirected graph with κκ components and independence number α(G)α(G), then there exist a positive integer k∈Nk∈N and a function f:VG→N0f:VG→N0 with non-negative integer values such that f(u)≤dG(u)f(u)≤dG(u) for u∈VGu∈VG, α(G)≥k≥∑u∈VG1dG(u)+1−f(u), and ∑u∈VGf(u)≥2(k−κ)∑u∈VGf(u)≥2(k−κ). This result is a best possible improvement of a result due to Harant and Schiermeyer [J. Harant, I. Schiermeyer, On the independence number of a graph in terms of order and size, Discrete Math. 232 (2001) 131–138] and implies that α(G)n(G)≥2(d(G)+1+2n(G))+(d(G)+1+2n(G))2−8 for connected graphs GG of order n(G)n(G), average degree d(G)d(G), and independence number α(G)α(G).