Local boxicity and maximum degree

Abstract

The local boxicity of a graph G, denoted by lbox(G), is the minimum positive integer l such that G can be obtained using the intersection of k (where k≥l) interval graphs where each vertex of G appears as a non-universal vertex in at most l of these interval graphs. Let G be a graph on n vertices having m edges. Let Δ denote the maximum degree of a vertex in G. We show that,•lbox(G)≤213log⁎⁡ΔΔ.•lbox(G)∈O(nlog⁡n).•lbox(G)≤(213log⁎⁡m+2)m.•the local boxicity of G is at most its product dimension. This connection helps us in showing that the local boxicity of the Kneser graphK(n,k) is at most k2log⁡log⁡n. The above results can be extended to the local dimension of a partially ordered set due to the known connection between local boxicity and local dimension. Using this connection along with known results it can be shown that there exist graphs of maximum degree Δ having a local boxicity of Ω(Δlog⁡Δ). There also exist graphs on n vertices and graphs on m edges having local boxicity of Ω(nlog⁡n) and Ω(mlog⁡m), respectively.