\newpage\section{Robertson}\label{robertson} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<–––––––––––––––––––– Robertson –––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grRobertson}{\oarg{options}\var{$k$}} \medskip From MathWord : \url{http://mathworld.wolfram.com/RobertsonGraph.html} \medskip \emph{The Robertson graph is the unique (4,5)-cage graph, illustrated below. It has 19 vertices and 38 edges. It has girth 5, diameter 3, chromatic number 3, and is a quartic graph.} \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{Robertson graph with \tkzcname{grRobertson} }} The cage \medskip \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[scale=.6] \GraphInit[vstyle=Art] \SetGraphArtColor{black}{gray} \grRobertson[RA=7] \end{tikzpicture} \end{tkzexample} \end{center} \clearpage\newpage \subsection{\tkzname{Fine embedding of the Robertson graph from RV}} \begin{center} \begin{tikzpicture}[scale=.8] \tikzstyle{TempEdgeStyle}= [thick,black,% double = gray,% double distance = 1.5pt]% \SetVertexNoLabel \renewcommand*{\VertexBigMinSize}{10pt} \GraphInit[vstyle=Shade] \SetVertexNoLabel \SetUpEdge[style = {thick,% double = orange,% double distance = 1pt}] \SetGraphShadeColor{gray}{black}{gray} \tikzstyle{EdgeStyle} = [TempEdgeStyle] \begin{scope}[rotate=-30] \grEmptyCycle[RA=5.4]{3} \end{scope} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend right=10] \grCycle[prefix=b,RA=4]{12} \tikzstyle{EdgeStyle}= [TempEdgeStyle] \grCirculant[prefix=c,RA=2]{4}{2} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend left] \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{4}{3}{4} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend right] \EdgeDoubleMod{c}{4}{0}{1} {b}{12}{8}{3}{4} \tikzstyle{EdgeStyle}= [TempEdgeStyle] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{11}{4}{3} \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{0}{3}{4} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend left=60] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{8}{4}{3} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend right=60] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{2}{4}{3} \tikzstyle{EdgeStyle}=[TempEdgeStyle,in=-50,out=-120, relative,looseness=2.5] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{5}{4}{3} \end{tikzpicture} \end{center} \clearpage\newpage Code for the Robertson Graph \medskip \begin{tkzexample}[code only] \begin{tikzpicture}[scale=.9] \tikzstyle{TempEdgeStyle}= [thick,black,% double = gray,% double distance = 1.5pt]% \SetVertexNoLabel \renewcommand*{\VertexBigMinSize}{14pt} \GraphInit[vstyle=Shade] \SetVertexNoLabel \SetUpEdge[style = {thick,% double = orange,% double distance = 1pt}] \SetGraphShadeColor{gray}{black}{gray} \tikzstyle{EdgeStyle} = [TempEdgeStyle] \begin{scope}[rotate=-30] \grEmptyCycle[RA=5.4]{3} \end{scope} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend right=10] \grCycle[prefix=b,RA=4]{12} \tikzstyle{EdgeStyle}= [TempEdgeStyle] \grCirculant[prefix=c,RA=2]{4}{2} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend left] \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{4}{3}{4} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend right] \EdgeDoubleMod{c}{4}{0}{1} {b}{12}{8}{3}{4} \tikzstyle{EdgeStyle}= [TempEdgeStyle] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{11}{4}{3} \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{0}{3}{4} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend left=60] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{8}{4}{3} \tikzstyle{EdgeStyle}= [TempEdgeStyle,bend right=60] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{2}{4}{3} \tikzstyle{EdgeStyle}=[TempEdgeStyle,in=-50,out=-120, relative,looseness=2.5] \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{5}{4}{3} \end{tikzpicture} \end{tkzexample} \clearpage\newpage \subsection{\tkzname{Robertson graph with new styles}} The code with new styles, the result is on the next page. \bigskip \begin{tkzexample}[code only] \begin{tikzpicture}[scale=1] \GraphInit[vstyle=Art] \SetGraphArtColor{gray}{red} \begin{scope}[rotate=-30] \grEmptyCycle[RA=5]{3} \end{scope} {\tikzset{EdgeStyle/.append style = {bend right=10}} \grCycle[prefix=b,RA=3.5]{12}} \grCirculant[prefix=c,RA=2]{4}{2} {\tikzset{EdgeStyle/.append style = {bend left}} \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{4}{3}{4}} {\tikzset{EdgeStyle/.append style = {bend right}} \EdgeDoubleMod{c}{4}{0}{1} {b}{12}{8}{3}{4}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{11}{4}{3} \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{0}{3}{4} {\tikzset{EdgeStyle/.append style = {bend left=60}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{8}{4}{3}} {\tikzset{EdgeStyle/.append style = {bend right=60}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{2}{4}{3}} {\tikzset{EdgeStyle/.append style = {in=-50,out=-120,% relative,looseness=2.5}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{5}{4}{3}} \end{tikzpicture} \end{tkzexample} \begin{center} \begin{tikzpicture}[scale=1] \GraphInit[vstyle=Art] \SetGraphArtColor{gray}{red} \begin{scope}[rotate=-30] \grEmptyCycle[RA=5]{3} \end{scope} {\tikzset{EdgeStyle/.append style = {bend right=10}} \grCycle[prefix=b,RA=3.5]{12}} \grCirculant[prefix=c,RA=2]{4}{2} {\tikzset{EdgeStyle/.append style = {bend left}} \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{4}{3}{4}} {\tikzset{EdgeStyle/.append style = {bend right}} \EdgeDoubleMod{c}{4}{0}{1} {b}{12}{8}{3}{4}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{11}{4}{3} \EdgeDoubleMod{c}{4}{0}{1}% {b}{12}{0}{3}{4} {\tikzset{EdgeStyle/.append style = {bend left=60}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{8}{4}{3}} {\tikzset{EdgeStyle/.append style = {bend right=60}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{2}{4}{3}} {\tikzset{EdgeStyle/.append style = {in=-50,out=-120,% relative,looseness=2.5}} \EdgeDoubleMod{a}{3}{0}{1}% {b}{12}{5}{4}{3}} \end{tikzpicture} \end{center} \clearpage\newpage %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<–––––––––––––––––––– Robertson Wegner –––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grRobertsonWegner}{\oarg{options}\var{$k$}} \medskip From MathWord : \url{http://mathworld.wolfram.com/Robertson-WegnerGraph.html} \medskip \emph{he Robertson-Wegner graph is of the four (5,5)-cage graphs, also called Robertson's cage . Like the other (5,5)-cages, the Robertson-Wegner graph has 30 nodes. It has 75 edges, girth 5, diameter 3, and chromatic number 4.} \href{http://mathworld.wolfram.com/Robertson-WegnerGraph.html}% {\textcolor{blue}{MathWorld}} by \href{http://en.wikipedia.org/wiki/Eric_W._Weisstein}% {\textcolor{blue}{E.Weisstein}} \end{NewMacroBox} \subsection{\tkzname{Robertson-Wegner graph}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[rotate=90,scale=.6] \GraphInit[vstyle=Art] \tikzset{VertexStyle/.append style={minimum size=2pt}} \grRobertsonWegner[RA=6] \end{tikzpicture} \end{tkzexample} \end{center} The next code gives the same result \begin{tkzexample}[code only] \begin{tikzpicture}[rotate=90] \GraphInit[vstyle=Art] \grLCF[RA=6]{6,12}{15} \EdgeInGraphMod{a}{30}{9}{1}{6} \EdgeInGraphMod*{a}{30}{15}{2}{6} \EdgeInGraphMod*{a}{30}{9}{3}{6} \end{tikzpicture} \end{tkzexample} \endinput