Boxicity and maximum degree

An axis-parallel d–dimensional box is a Cartesian product R1 × R2 × · · · × Rd where Ri (for 1 ≤ i ≤ d) 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 d, such that G is representable as the intersection graph of (axis–parallel) boxes in d–dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc. We show that for any graph G with maximum degree ∆, box(G) ≤ 2∆ + 2. That the bound does not depend on the number of vertices is a bit surprising considering the fact that there are highly connected bounded degree graphs such as expander graphs. Our proof is very short and constructive. We conjecture that box(G) is O(∆). Let F = {Sx ⊆ U : x ∈ V } be a family of subsets of a universe U , where V is an index set. The intersection graph Ω(F) of F has V as vertex set, and two distinct vertices x and y are adjacent if and only if Sx ∩ Sy 6= ∅. Representation of graphs as the intersection graphs of various geometrical objects is a well studied topic in graph theory. A prime example of a graph class defined in this way is the class of interval graphs: A graph G is an interval graph if and only if G has an interval realization: i.e., each vertex of G can be associated to an interval on the real line such that two intervals intersect if and only if the corresponding vertices are adjacent. Motivated by theoretical as well as practical considerations, graph theorists have tried to generalize the concept of interval graphs in various ways. One such generalization is the concept of boxicity defined as follows. An axis-parallel d–dimensional box is a Cartesian product R1 × R2 × · · · × Rd where Ri (for 1 ≤ i ≤ d) 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 d, such that G is representable as the intersection graph of (axis–parallel) boxes in d–dimensional space. It is easy to see that the class of graphs with d ≤ 1 is exactly the class of Preprint submitted to Elsevier Science