Lower Bounds for Boxicity [article]

Abhijin Adiga, L. Sunil Chandran, Naveen Sivadasan
2012 arXiv   pre-print
An axis-parallel b-dimensional box is a Cartesian product R_1× R_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.
more » ... 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- 40 n/n^2. 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). (3) Spectral lower bounds for the boxicity of k-regular graphs. (4) The boxicity of random k-regular graphs on n vertices (where k is fixed) is Ω(k/ k). (5) There exists a positive constantc such that almost all balanced bipartite graphs on 2n vertices with exactly m edges have boxicity at least c m/n, for m< c' n^2/3 for any positive constant c' < 1.
arXiv:0806.3175v2 fatcat:lypycwxyhzapheqdqsqbzz73bq