PerspectiveBoxicity 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 Rk. In this paper we show that for a line graph G of a multigraph, box(G)≤2Δ(G)(⌈log2log2Δ(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)log2log2(χ(G))). For the d-dimensional hypercube Qd, we prove that box(Qd)≥12(⌈log2log2d⌉+1). The question of finding a nontrivial lower bound for box(Qd) 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].The above 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).

► Boxicity of a line graph G is O(D(loglogD)), where D denotes the maximum degree of G. ► Boxicity of a line graph G is O(k(loglogk)), where k denotes the chromatic number of G. ► Boxicity of the d-dimensional hypercube is at least (1/2)(⌈loglogd⌉+1).