Lower bounds for boxicity

An axis-parallel b-dimensional box is a Cartesian product R1×R2×· · ·×Rb where Ri is a closed interval of the form [ai, bi] 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 be such that c1/n ≤ p ≤ 1− c2 log n n2 , where c1 and c2 are predetermined constants. Then, for G ∈ G(n, p), almost surely 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: Let m be an integer such that c1n ≤ m ≤ c3n 2 (c3 is a suitable constant). Then, there exists a constant c4 such that almost all graphs on n vertices and exactly m edges have boxicity at least c4m/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 „ k2/λ2 log(1+k2/λ2) «