Independence in connected graphs

We prove that if G = (VG, EG) is a finite, simple, and undirected graph with κ components and independence number α(G), then there exist a positive integer k ∈ N and a function f : VG → N0 with non-negative integer values such that f(u) ≤ dG(u) for u ∈ VG, α(G) ≥ k ≥ ∑ u∈VG 1 dG(u)+1−f(u) , and ∑ u∈VG f(u) ≥ 2(k − κ). This result is a best-possible improvement of a result due to Harant and 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+ 2 n(G) ) + √( d(G)+1+ 2 n(G) )2 −8 for connected graphs G of order n(G), average degree d(G), and independence number α(G).