Boxicity of graphs with bounded degree

The boxicity of a graph G=(V,E)G=(V,E) is the smallest kk for which there exist kk interval graphs Gi=(V,Ei)Gi=(V,Ei), 1≤i≤k1≤i≤k, such that E=E1∩…∩EkE=E1∩…∩Ek. Graphs with boxicity at most dd are exactly the intersection graphs of (axis-parallel) boxes in RdRd. In this note, we prove that graphs with maximum degree ΔΔ have boxicity at most Δ2+2Δ2+2, which improves the previous bound of 2Δ22Δ2 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. http://dx.doi.org/10.1016/j.jctb.2007.08.002].