The independence number in graphs of maximum degree three

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We prove that a K 4 -free graph G of order n , size m and maximum degree at most three has an independent set of cardinality at least 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].