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).

• Type: Journal
• Year: 2011
• Where: DAM
• Authors: Jochen Harant, Dieter Rautenbach
• Keywords: DAM 2011, Independence Number, Integer, Non-negative Integer, Theoretical Computer Science