Local boxicity and maximum degree

Majumder, Atrayee and Mathew, Rogers (2022) Local boxicity and maximum degree. Discrete Mathematics, 345 (12). ISSN 0012-365X

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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)≤213logΔ. • lbox(G)∈O([Formula presented]). • lbox(G)≤(213log+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 graph K(n,k) is at most [Formula presented]log⁡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([Formula presented]). There also exist graphs on n vertices and graphs on m edges having local boxicity of Ω([Formula presented]) and Ω([Formula presented]), respectively. © 2022 Elsevier B.V.

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