\newpage\section{Heawood graph}\label{heawood} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––– HEAWOOD ––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grHeawood}{\oarg{options}} \medskip From Wikipedia \url{http://en.wikipedia.org/wiki/Heawood_graph} \emph{The Heawood graph is an undirected graph with 14 vertices and 21 edges. Each vertex is adjacent to exactly three edges (that is, it is a cubic graph), and all cycles in the graph have six or more edges. Percy John Heawood (1861-1955) was an English mathematician who spent a large amount of time on questions related to the four colour theorem.} \medskip From MathWorld \url{http://mathworld.wolfram.com/HeawoodGraph.html} \emph{The Heawood graph is the unique $(3,6)$-cage graph and Moore graph and is graph illustrated below in one of his embeddings.} \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}} \end{NewMacroBox} \subsection{\tkzname{Heawood graph}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}% \GraphInit[vstyle=Shade] \grHeawood[RA=7] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage It can be represented in LCF notation as $\big[5,-5\big]^7$ \tkzcname{grLCF[RA=5]\{5,9\}\{7\}} gives the result because $-5 = 9\ mod\ 14$. \subsection{\tkzname{Heawood graph with LCF notation}}\label{lcf2} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}% \GraphInit[vstyle=Art] \grLCF[RA=7]{5,9}{7}% \end{tikzpicture} \end{tkzexample} \end{center} \vfill\endinput