Independent sets in graphs with triangles

Abstract

In this note we give a fast algorithm which computes an independent set of size at least Ω(( n Δ ) ln Δ) for a graph G on n vertices with maximum degree Δ, if G contains only a little less than the maximum possible number of triangles (say nΔ 2 − ε many for a positive constant ε). Our algorithm uses derandomization and is based on earlier results concerning the independence number of triangle-free graphs due to Ajtai et al. (1980) and Shearer (1983).