A Cayley graph

The following is the Cayley graph , where is the additive group of the vector space (where is the field of three elements), and .

\begin{tikzpicture}
  \tikzset{VertexStyle/.style={draw,rectangle}}
  \grEmptyCycle*[rotation=90,prefix=a,RA=2,Math]{{(1,2)},{(0,0)},{(2,1)}}
  \grEmptyCycle*[rotation=90,prefix=b,RA=4.5,Math]{(2,0),(1,0),(1,1),(0,1),(0,2),(2,2)}
  \EdgeInGraphLoop{b}{6}
  \SetUpEdge[style={bend right=10}]
  \EdgeDoubleMod{a}{3}{0}{1}{b}{6}{1}{2}{3}
  \EdgeDoubleMod{a}{3}{0}{1}{b}{6}{2}{2}{3}
  \SetUpEdge[style={bend left=10}]
  \EdgeDoubleMod{a}{3}{0}{1}{b}{6}{4}{2}{3}
  \EdgeDoubleMod{a}{3}{0}{1}{b}{6}{5}{2}{3}    
\end{tikzpicture}

Introducing graphiso.sty

The purpose of the new package graphiso.sty is to use tkz-graph to produce an animation of a graph isomorphism in a beamer presentation, such as:

You can find documentation and code at Github: rvf0068/graphiso.sty.

An icosahedron in fake 3d

\begin{tikzpicture}
  \begin{scope}[rotate=90]
    \grIcosahedral[form=1,RA=3,RB=1.5]
    \SetUpEdge[color=white,style={double=black,double distance=2pt}]
    \EdgeInGraphLoop{a}{6}
    \EdgeFromOneToSel{a}{b}{0}{1,5}
    \Edges(a2,b1,b3,b5,a4)
    \Edge(a3)(b3)
    \Edges(a1,b1,b5,a5)
    \Edges(a2,b3,a4)
  \end{scope}
\end{tikzpicture}

Options for vertices

This post is an update for this post from my old blog, which did not work with the newer versions of tkz-berge. The changes are:

• instead of \tikzstyle{every node} = [node distance=1.5cm] one now has to use the macro: \SetGraphUnit{1.5},
• the macro \Vertices needs now an extra argument: {line} (another useful option here is {circle})
• the macro \Edges in line 16 needs the extra option local, in order for the other options (the color red) to work.

\begin{tikzpicture}
  \SetVertexNormal[LineColor=brown]
  \Vertex[x=0,y=2.5,style=orange,LabelOut=true,Lpos=90]{A}
  {\SetGraphUnit{1.5}
    \Vertices[x=1.5,y=4,dir=\SO,LabelOut=true,Ldist=5pt]{line}{B,C,D}}
  \Vertices[x=3,y=5,dir=\SO,style={shape=coordinate}]{line}{E,F,G,H,I,J}
  \Vertices[x=4.5,y=5,dir=\SO,style={font=\bfseries}]{line}{K,L,M,N,O,P}
  {\SetGraphUnit{1.5}
    \Vertices[x=6,y=4,dir=\SO,
    style={line width=2pt,
      inner sep=0pt,
      text=purple,
      fill=yellow,
      minimum size=12pt}]{line}{Q,R,S}}
  \Vertex[x=7.5,y=2.5,style={shape=rectangle,blue}]{T}
  \Edges[local,color=red,lw=2pt](A,B,E,K,Q,T,S,P,J,D,A)
  \Edges(B,G,L,R,T)
  \Edges(D,H,O,R)
  \Edges(A,C,F,M,Q)
  \Edges(C,I,N,S)
  \foreach \x/\y in {E/F,G/H,I/J,K/L,M/N,O/P}
  {\Edge(\x)(\y)}
  \Edge[style={bend left}](G)(N)
  \Edge[style={bend right}](A)(T)
\end{tikzpicture}

UPDATE For some reason I do not understand, the code above does not work with an updated installation of TeX Live 2011. (It worked for me with the TeX Live 2009 distributed in Ubuntu repositories). The following code produces the intended picture. Until it is figured out why the options to the \Vertex macro do not work well, it seems to be more predictable to use commands instead of options to the macro.

Such commands are defined starting at line 72 of tkz-graph.sty.

\begin{tikzpicture}
  \SetVertexNormal[LineColor=brown]
  {\renewcommand{\VertexLightFillColor}{orange}
    \Vertex[x=0,y=2.5,style=orange,LabelOut=true,Lpos=90]{A}}
  {\SetGraphUnit{1.5}
    \Vertices[x=1.5,y=4,dir=\SO,LabelOut=true,Ldist=5pt]{line}{B,C,D}}
  {\renewcommand{\VertexShape}{coordinate}
    \Vertices[x=3,y=5,dir=\SO]{line}{E,F,G,H,I,J}}
  \Vertices[x=4.5,y=5,dir=\SO,style={font=\bfseries}]{line}{K,L,M,N,O,P}
  {\SetGraphUnit{1.5}
    \renewcommand{\VertexLineWidth}{2pt}
    \renewcommand{\VertexInnerSep}{0pt}
    \renewcommand{\VertexTextColor}{purple}
    \renewcommand{\VertexLightFillColor}{yellow}
    \renewcommand{\VertexInterMinSize}{12pt}
    \Vertices[x=6,y=4,dir=\SO]{line}{Q,R,S}}
  {\renewcommand{\VertexShape}{rectangle}
    \renewcommand{\VertexLightFillColor}{blue}
    \Vertex[x=7.5,y=2.5]{T}}
  \Edges[local,color=red,lw=2pt](A,B,E,K,Q,T,S,P,J,D,A)
  \Edges(B,G,L,R,T)
  \Edges(D,H,O,R)
  \Edges(A,C,F,M,Q)
  \Edges(C,I,N,S)
  \foreach \x/\y in {E/F,G/H,I/J,K/L,M/N,O/P}
  {\Edge(\x)(\y)}
  \Edge[style={bend left}](G)(N)
  \Edge[style={bend right}](A)(T)
\end{tikzpicture}

Edges on a grid

For drawing some extra edges in a grid I used the following code:

\begin{tikzpicture}
  \newcounter{xp}
  \newcounter{yp}
  \grGrid[prefix=a,RA=2,RB=2]{6}{6}
  \foreach \x in {0,2,4}{%
    \foreach \y in {1,3}{%
      \setcounter{xp}{\x}
      \setcounter{yp}{\y}
      \stepcounter{xp}
      \stepcounter{yp}
      \Edge(a\x;\y)(a\thexp;\theyp)
      }
    }
\end{tikzpicture}

For some reason the following code does not work (producing the error ! Package pgf Error: No shape named a1 is known.) Apparently, only counters are permitted as vertex indices.

\begin{tikzpicture}
  \grGrid[prefix=a,RA=2,RB=2]{6}{6}
  \foreach \x in {0,2,4}{%
    \foreach \y in {1,3}{%
      \pgfmathsetmacro{\xp}{\x+1}
      \pgfmathsetmacro{\yp}{\y+1}
      \Edge(a\x;\y)(a\xp;\yp)
      }
    }
\end{tikzpicture}

The flower snarks

I’m using the new rotation option for cycles to define a macro that draws a flower snark. The macro has as optional arguments the radius RA, RB, and RC of the three types of vertices, and a mandatory argument which must be an odd natural number.

\newcommand{\grFlowerSnark}[2][]{%
  \begingroup
  \setkeys[GR]{cl}{#1}%
  \edef\tkzb@rtemp{\cmdGR@cl@RA}
  \edef\tkzb@rtempx{\cmdGR@cl@RB}
  \edef\tkzb@ptemp{\cmdGR@cl@RC}
  \pgfmathsetcounter{tkz@gr@a}{(#2-1)/2}
  \pgfmathsetcounter{tkz@gr@b}{(#2+1)/2}
  \grCycle[RA=\tkzb@ptemp,prefix=b,rotation=90]{#2}
  \grEmptyCycle[RA=\tkzb@rtempx,prefix=a,rotation=90]{#2}
  \pgfmathsetmacro{\smallshift}{360/(5*#2)}
  \grEmptyCycle[RA=\tkzb@rtemp,prefix=c,rotation=90+\smallshift]{#2}
  \grEmptyCycle[RA=\tkzb@rtemp,prefix=d,rotation=90-\smallshift]{#2}
  \EdgeIdentity{a}{b}{#2}
  \EdgeIdentity{a}{c}{#2}
  \EdgeIdentity{a}{d}{#2}
  \Edge(c\thetkz@gr@a)(d\thetkz@gr@b)
  \pgfmathsetcounter{tkz@gr@c}{\thetkz@gr@a-1}
  \pgfmathsetcounter{tkz@gr@d}{\thetkz@gr@b+1}
  {\tikzset{EdgeStyle/.append style = {bend right=90}}
    \EdgeDoubleMod{c}{#2}{0}{1}{c}{#2}{1}{1}{\thetkz@gr@c}
    \EdgeDoubleMod{c}{#2}{\thetkz@gr@b}{1}{c}{#2}{\thetkz@gr@d}{1}{\thetkz@gr@c}
    \EdgeDoubleMod{d}{#2}{0}{1}{d}{#2}{1}{1}{\thetkz@gr@c}
    \EdgeDoubleMod{d}{#2}{\thetkz@gr@b}{1}{d}{#2}{\thetkz@gr@d}{1}{\thetkz@gr@c}
  }
  \Edge[style={bend right=90}](d\thetkz@gr@a)(c\thetkz@gr@b)
  \endgroup
}

So that, for example:

\begin{tikzpicture}
  \SetVertexNormal[MinSize=10pt]
  \grFlowerSnark[RA=4,RB=2.5,RC=1]{5}
\end{tikzpicture}

gives:

The Szekeres snark, more new features

\grEmptyCycle and \grCycle can also have options x and y, that set the coordinates of the center of the cycle, or r and d for polar coordinates (r is the radius and d the angle). Here we use these options together with rotation, in order to draw the Szekeres snark.

\newcounter{in}
\newcounter{nex}
\newcounter{ney}
\begin{tikzpicture}[rotate=90,scale=0.8]
  \SetUpVertex[Style={font=\footnotesize\sffamily}]
  \SetVertexNormal[MinSize=12pt,InnerSep=0pt]
  \grEmptyCycle[prefix=x]{5}
  \foreach \x in  {0,1,...,4}{%
    \setcounter{in}{\x}
    \stepcounter{in}
    \pgfmathsetmacro{\rad}{72*\x}
    \grEmptyCycle[RA=1.65,r=4,d=\rad,prefix=\alph{in},rotation=\rad-160]{9}
    \EdgeInGraphLoop*{\alph{in}}{9}
    \Edge(\alph{in}0)(\alph{in}5)
    \Edge(\alph{in}8)(\alph{in}3)
    \foreach \y in {1,4,7}{%
      \Edge(x\x)(\alph{in}\y)}
  }
  \foreach  \x in  {0,1,...,4}{%
    \setcounter{in}{\x}
    \stepcounter{in}
    \pgfmathsetcounter{nex}{mod(\x+1,5)+1}
    \setcounter{ney}{\thenex}
    \Edge(\alph{in}0)(\alph{ney}8)
    \pgfmathsetcounter{nex}{mod(\x+2,5)+1}
    \setcounter{ney}{\thenex}
    \Edge(\alph{in}6)(\alph{ney}2)
  }
\end{tikzpicture}