Geometric representation of graphs in low dimension

Abstract

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5(\Delta +2)\ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (Δ+ 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot (d_{av}+1)\ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b\sqrt{\ln n}$ for a constant b.

preliminary version appeared in Cocoon 2006