Local Boxicity and Maximum Degree

TL;DR

This paper introduces the concept of local boxicity, establishing bounds related to maximum degree, number of edges, and vertices, and explores its connections to product dimension, local dimension, and cubicity in graphs.

Contribution

It defines local boxicity and derives upper and lower bounds based on graph parameters, connecting it to other graph dimensions and extending results to specific graph classes.

Findings

Bound on local boxicity in terms of maximum degree and iterated logarithm.

Existence of graphs with high local boxicity relative to degree, edges, and vertices.

Connection between local boxicity and product dimension, with applications to Kneser graphs and cubicity.

Abstract

The \emph{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 $k \geq l$,) 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 $\Delta$ denote the maximum degree of a vertex in $G$. We show that, (i) $lbox(G) \leq 2^{13\log^{*}{\Delta}} \Delta$. There exist graphs of maximum degree $\Delta$ having a local boxicity of $\Omega(\frac{\Delta}{\log\Delta})$. (ii) $lbox(G) \in O(\frac{n}{\log{n}})$. There exist graphs on $n$ vertices having a local boxicity of $\Omega(\frac{n}{\log n})$. (iii) $lbox(G) \leq (2^{13\log^{*}{\sqrt{m}}} + 2 )\sqrt{m}$. There exist graphs with $m$ edges having a local boxicity of $\Omega(\frac{\sqrt{m}}{\log m})$. (iv) the local boxicity of $G$ is at most its \emph{product dimension}. This connection helps us in showing that the local boxicity of the \emph{Kneser graph} $K(n,k)$ is at most $\frac{k}{2} \log{\log{n}}$. The above results can be extended to the \emph{local dimension} of a partially ordered set due to the known connection between local boxicity and local dimension. Finally, we show that the \emph{cubicity} of a graph on $n$ vertices of girth greater than $g+1$ is $O(n^{\frac{1}{\lfloor g/2\rfloor}}\log n)$.