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arXiv · 1208.4734

New approach to the $k$-independence number of a graph

Yair Caro·Adriana Hansberg
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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 $α_k(G)$ 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 $d$, $α_k(G) \ge \frac{k+1}{\lceil d \rceil + k + 1} n$, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99-107].

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Yair Caro, Adriana Hansberg. 2012-08-23. New approach to the $k$-independence number of a graph. https://arxiv.org/abs/1208.4734

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