On the boxicity of Kneser graphs and complements of line graphs

Abstract

The boxicity of a graph G=(V,E), denoted by box(G), is the minimum dimension d such that G is the intersection graph of a family (Bv)v∈V of d-dimensional axis-parallel boxes in Rd.Let k and n be two positive integers such that n≥2k+1. The Kneser graphK(n,k) is the graph with vertex set given by all subsets of {1,2,…,n} of size k where two vertices are adjacent if their corresponding k-sets are disjoint. In this note, we derive a general upper bound for box(K(n,k)), and a lower bound in the case n≥2k3−2k2+1, which matches the upper bound up to an additive factor of Θ(k2). Our second contribution is to provide upper and lower bounds for the boxicity of the complement of the line graph of any graph G, and as a corollary we derive that box(K(n,2))∈{n−3,n−2} for every n≥5.