1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
|
\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
|