\newpage\section{Franklin}\label{franklin} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––– Franklin ––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grFranklin}{\oarg{options}} \medskip From MathWord : \url{http://mathworld.wolfram.com/FranklinGraph.html} \emph{The Franklin graph is the 12-vertex cubic graph shown above whose embedding on the Klein bottle divides it into regions having a minimal coloring using six colors, thus providing the sole counterexample to the Heawood conjecture.} \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}} \medskip The Franklin graph is implemented in \tkzname{tkz-berge} as \tkzcname{grFranklin}. \end{NewMacroBox} \tikzstyle{VertexStyle} = [shape = circle,% color = white, fill = black, very thin, inner sep = 0pt,% minimum size = 18pt, draw] \tikzstyle{EdgeStyle} = [thick,% double = brown,% double distance = 1pt] \newcounter{tempi}\setcounter{tempi}{0} \subsection{\tkzname{The Franklin graph : embedding 1}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[scale=.7] \grFranklin[Math,RA=7] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{The Franklin graph : embedding 2}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \grCycle[Math,RA=4,prefix=a]{6} \grCycle[Math,RA=6,prefix=b]{6} \foreach \x in {0,...,5}{% \ifthenelse{\isodd{\x}}{% \pgfmathsetcounter{tempi}{\x-1}}{% \pgfmathsetcounter{tempi}{\x+1}} \Edge(a\x)(b\thetempi) } \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{The Franklin graph : with LCF notation embedding 3}} \space*{2cm} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \grLCF[Math,RA=7]{-5,-3,3,5}{3} \end{tikzpicture} \end{tkzexample} \end{center} \endinput