Boxicity of graphs with bounded degree

Abstract

The boxicity of a graph G = ( V , E ) is the smallest k for which there exist k interval graphs G i = ( V , E i ) , 1 ≤ i ≤ k , such that E = E 1 ∩ … ∩ E k . Graphs with boxicity at most d are exactly the intersection graphs of (axis-parallel) boxes in R d . In this note, we prove that graphs with maximum degree Δ have boxicity at most Δ 2 + 2 , which improves the previous bound of 2 Δ 2 obtained by Chandran et al. [L.S. Chandran, M.C. Francis, N. Sivadasan, Boxicity and maximum degree, J. Combin. Theory Ser. B 98 (2008) 443–445. doi:10.1016/j.jctb.2007.08.002].