From 877268a0de707a979be934d888518f6cc02d73a6 Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Tue, 10 Mar 2020 03:01:27 +0000
Subject: CTAN sync 202003100301

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+\newpage\section{Levi Graph}\label{levi}
+%<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––>
+%<––––––––––––––––––––      Levy             –––––––––––––––––––––––––––––––>
+%<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––>
+\begin{NewMacroBox}{grLevi}{\oarg{options}}
+
+\medskip
+
+From Wikipedia \url{http://en.wikipedia.org/wiki/Levi_graph}
+
+\emph{In combinatorics a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line.\hfil\break 
+In the mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth 8 it is a cage and a Moore graph. It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle. }
+
+From MathWord : \url{http://mathworld.wolfram.com/LeviGraph.html} 
+
+\emph{It has 30 nodes and 45 edges. It has girth 8, diameter 4, chromatic number 2. The Levi graph is a generalized polygon which is the point/line incidence graph of the generalized quadrangle . The graph  was first discovered by Tutte (1947), and is also called the Tutte-Coxeter graph , Tutte's cage  or "Tutte's (3,8)-cage".The Levi graph is the unique (3,8)-cage graph.\hfil\break
+The incidence graph of a generic configuration is sometimes known as a Levi graph (Coxeter 1950).} 
+
+\href{http://mathworld.wolfram.com/topics/GraphTheory.html}%
+           {\textcolor{blue}{MathWorld}} by \href{http://en.wikipedia.org/wiki/Eric_W._Weisstein}%
+           {\textcolor{blue}{E.Weisstein}}   
+           
+Some examples of Levi Graphs with this definition are~: 
+\begin{itemize}
+\item Desargues graph
+\item Heawood graph
+\item Heawood graph
+\item Pappus graph
+\item Gray graph 
+\item Tutte eight-cage 
+\end{itemize}
+
+\end{NewMacroBox}
+
+The two forms can be draw with :
+
+ \begin{tkzexample}[code only]
+   \grLevi[RA=7]\end{tkzexample}
+
+and
+
+ \begin{tkzexample}[code only]
+    \grLevi[form=2,RA=7,RB=5,RC=3]\end{tkzexample}
+
+You can see on the next pages, the two forms.
+\vfill\newpage
+Now I show you how to code this graph.
+
+\subsection{\tkzname{Levy graph :form 1}}
+
+\bigskip
+\begin{center}
+\begin{tkzexample}[vbox]
+  \begin{tikzpicture}
+    \GraphInit[vstyle=Art]
+    \grLCF[prefix=a,RA=6]{-13,-9,7,-7,9,13}{5}
+ \end{tikzpicture}
+\end{tkzexample} 
+\end{center}
+
+
+\vfill\newpage
+
+\subsection{\tkzname{Levy graph :form 2}}
+
+\bigskip
+\begin{center}
+\begin{tkzexample}[vbox]
+\begin{tikzpicture}
+  \GraphInit[vstyle=Art]
+  \grCycle[prefix=a,RA=7]{10}
+  \EdgeInGraphMod{a}{10}{5}
+  \grEmptyCycle[prefix=b,RA=5]{10}
+  \grEmptyCycle[prefix=c,RA=3]{10}
+  \EdgeInGraphMod{c}{10}{4}
+ \end{tikzpicture}
+\end{tkzexample} 
+\end{center}
+\endinput
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