Boxicity of line graphs

Boxicity of a graph H , denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel kdimensional boxes in R. In this paper, we show that for a line graph G of a multigraph, box(G) ≤ 2∆(⌈log 2 log 2 ∆⌉ + 3) + 1, where ∆ denotes the maximum degree of G. Since ∆ ≤ 2(χ−1), for any line graph G with chromatic number χ, box(G) = O(χ log 2 log 2 (χ)). For the d-dimensional hypercube Hd, we prove that box(Hd) ≥ 1 2 (⌈log 2 log 2 d⌉+ 1). The question of finding a non-trivial lower bound for box(Hd) was left open by Chandran and Sivadasan in [L. Sunil Chandran and Naveen Sivadasan. The cubicity of Hypercube Graphs. Discrete Mathematics, 308(23):57955800, 2008]. The above results are consequences of bounds that we obtain for the boxicity of fully subdivided graphs (a graph which can be obtained by subdividing every edge of a graph exactly once).