Boxicity of Line Graphs
Abstract
Boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in Rk. In this paper, we show that for a line graph G of a multigraph, box(G) <= 2( log2(log2()) + 3) + 1, where denotes the maximum degree of G. Since <= 2( - 1), for any line graph G with chromatic number , box(G) = O( log2(log2())). For the d-dimensional hypercube Hd, we prove that box(Hd) >= ( log2(log2(d)) + 1)/2. 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):5795-5800, 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).