Sphere of Influence Dimension Conjecture 'Almost Proved'

英文摘要：

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 $$SIG(G)\leq \bigg{ \lfloor}\frac{2n}{3}\bigg{ \rfloor}+2.$$