Let G = (V,E) be a graph and k > 0 an integer. A k-independent set S V is a set of vertices such that the maximum degree in the graph induced by S is at most k. With k(G) we denote the maximum cardinality of a k-independent set of G. We prove that, for a graph G on n vertices and average degree d, k(G) > k+1 dde+k+1n, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99–107].
Caro, Y.; Hansberg, A. New approach to the k-independence number of a graph. "Electronic journal of combinatorics", 2013, vol. 20, núm. 1, p. 1-16.