The independence number in graphs of maximum degree three

We prove that a K₄-free graph G of order n, size m and maximum degree at most three has an independent set of cardinality at least frac(1, 7) (4 n - m - λ - t r), where λ counts the number of components of G whose blocks are each either isomorphic to one of four specific graphs or edges between two of these four specific graphs and tr is the maximum number of vertex-disjoint triangles in G. Our result generalizes a bound due to Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233-237].