Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local Boxicity and Maximum Degree   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 $O(n^{\frac{1}{\lfloor g/2\rfloor}}\log
n)$.

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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int