
Yair Caro

Adriana Hansberg
Keywords:
$k$independence, average degree
Abstract
Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ 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 $d$, $\alpha_k(G) \ge \frac{k+1}{\lceil d \rceil + k + 1} n$, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on $k$independence, J. Graph Theory 15 (1991), 99107].