An improved upper bound for the size of the sphere of influence graph

TL;DR

This paper improves the upper bound on the number of edges in the sphere of influence graph for points in the plane, reducing the constant from 15 to 14.5, thus advancing understanding of its combinatorial properties.

Contribution

The paper establishes a tighter upper bound of 14.5 on the constant for the maximum edges in the sphere of influence graph, improving previous bounds.

Findings

Proves the upper bound c=14.5 for the sphere of influence graph

Refines previous bounds from 15 to 14.5

Contributes to combinatorial geometry theory

Abstract

Let $V$ be a set of $n$ points in the plane. For each $x\in V$, let $B_x$ be the closed circular disk centered at $x$ with radius equal to the distance from $x$ to its closest neighbor. The {\it closed sphere of influence graph} on $V$ is defined as the undirected graph where $x$ and $y$ are adjacent if and only if the $B_x$ and $B_y$ have nonempty intersection. It is known that every $n$-vertex closed sphere of influence graph has at most $cn$ edges, for some absolute positive constant $c$. The first result was obtained in 1985 by Avis and Horton who provided the value $c=29$. Their result was successively improved by several authors: Bateman and Erd\H{o}s (c=18), Michael and Quint (c=17.5), and Soss (c=15). In this paper we prove that one can take $c=14.5$.