Sphere of Influence Dimension Conjecture 'Almost Proved'

TL;DR

This paper makes significant progress towards the Sphere of Influence Dimension Conjecture by proving an upper bound that closely approaches the conjectured limit, advancing understanding of geometric graph representations.

Contribution

The paper nearly proves the SIG dimension conjecture by establishing an upper bound within 2 of the conjectured value, improving previous bounds.

Findings

Established an upper bound for SIG(G) as ⌊2n/3⌋+2.

Progressed towards proving the SIG dimension conjecture.

Provided new insights into sphere-of-influence graph representations.

Abstract

The sphere-of-influence graph (SIG) on a finite set of points in a metric space, each with an open ball centred about it of radius equal to the distance between that point and its nearest neighbor, is defined to be the intersection graph of these balls. Let $G$ be a graph of order $n,$ having no isolated vertices. The SIG-dimension of $G,$ denoted by $SIG(G),$ is defined to be the least possible $d$ such that $G$ can be realized as a sphere of influence graph in $\mathbb{R}^d,$ equipped with sup-norm. In 2000, Boyer [E. Boyer, L. Lister and B. Shader, Sphere of influence graphs using the sup-norm, Mathematical and Computer Modelling 32 (2000) 1071-1082] put forward the SIG dimension conjecture, which states that $$SIG(G)\leq \bigg\lceil \frac{2n}{3}\bigg\rceil.$$ In this paper, we 'almost' establish this conjecture by proving that $$SIG(G)\leq \bigg{ \lfloor}\frac{2n}{3}\bigg{ \rfloor}+2.$$