Lower bounds for boxicity

Abstract

An axis-parallel b-dimensional box is a Cartesian product R ₁×–₂×...×R _b where R _i is a closed interval of the form [a _i ; b _i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below:

• 1. The boxicity of a graph on n vertices with no universal vertices and minimum degree δ is at least n/2(n−δ−1).

• 2. Consider the G(n;p) model of random graphs. Let p ≤ 1 − 40logn/n ². Then with high probability, box(G) = Ω(np(1 − p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Ω(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and $m \leqslant c\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)$ edges have boxicity Ω(m/n).

• 3. Let G be a connected k-regular graph on n vertices. Let λ be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is at least $\left( {\frac{{k^2 /\lambda ^2 }} {{\log \left( {1 + k^2 /\lambda ^2 } \right)}}} \right)\left( {\frac{{n - k - 1}} {{2n}}} \right)$.

• 4. For any positive constant c < 1, almost all balanced bipartite graphs on 2n vertices and m≤cn ² edges have boxicity Ω(m/n).