How large part of a graph can be covered by the neighborhoods of k vertices?

TL;DR

This paper proves that in a graph with average degree proportional to the number of vertices, there exist k vertices whose neighborhoods cover a significant portion of the graph, with bounds that are nearly optimal.

Contribution

It establishes a tight bound on the maximum coverage of vertices by neighborhoods of k vertices in such graphs, answering a question posed by Simon Griffiths.

Findings

Existence of k vertices with neighborhoods covering at least a specific fraction of the graph.

The coverage bound is tight and nearly optimal.

Provides a mathematical answer to an open question in graph theory.

Abstract

Let $k\ge 2$ be fixed integer, $0<c<1$ a constant. Consider a graph $G$ with $n$ vertices and average degree $cn$. We answer a question of Simon Griffiths by showing that $G$ has $k$ vertices such that their neighborhoods together cover at least $\min(1-(1-c)^{k},\sqrt{c})n$ vertices. This result is essentially tight.