Boxicity of line graphs

The 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 R k . In this paper we show that for a line graph G of a multigraph, box ( G ) ≤ 2 Δ ( G ) ( ⌈ log 2 log 2 Δ ( G ) ⌉ + 3 ) + 1 , where Δ ( G ) denotes the maximum degree of G . Since G is a line graph, Δ ( G ) ≤ 2 ( χ ( G ) − 1 ) , where χ ( G ) denotes the chromatic number of G , and therefore, box ( G ) = O ( χ ( G ) log 2 log 2 ( χ ( G ) ) ) . For the d -dimensional hypercube Q d , we prove that box ( Q d ) ≥ 1 2 ( ⌈ log 2 log 2 d ⌉ + 1 ) . The question of finding a nontrivial lower bound for box ( Q d ) was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800]. ove results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once).