Interpolating between bounds on the independence number

For a non-negative integer T, we prove that the independence number of a graph G = (V, E) in which every vertex belongs to at most T triangles is at least uV f(d(u), T) where d(u) denotes the degree of a vertex u V , f(d, T) = 1 d+1 for T d 2 and f(d, T) = (1 + (d2 - d - 2T)f(d - 1, T))/(d2 + 1 - 2T) for T < d 2 . This is a common generalization of the lower bounds for the independence number due to Caro, Wei, and Shearer. We discuss further possible strengthenings of our result and pose a corresponding conjecture.

• Type: Journal
• Year: 2010
• Where: DM
• Authors: Anett Boßecker, Dieter Rautenbach
• Keywords: DM 2010, Independence Number, Non-negative Integer, Triangles