%<–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \newpage\section{Balaban}\label{balaban} %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> %<–––––––––––––––––––––– Balaban's graph ––––––––––––––––––––––––––––––––> %<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{NewMacroBox}{grBalaban}{\oarg{options}} \medskip From MathWord : \url{http://mathworld.wolfram.com/Balaban10-Cage.html} \emph{The Balaban 10-cage is one of the three(3,10)-cage graphs (Read 1998, p. 272). The Balaban (3,10)-cage was the first known example of a 10-cage (Balaban 1973; Pisanski 2001). Embeddings of all three possible (3,10)-cages (the others being the Harries graph and Harries-Wong graph) are given by Pisanski et al. (2001). Several embeddings are illustrated below, with the three rightmost being given by Pisanski and Randić (2000) It is a Hamiltonian graph and has Hamiltonian cycles. It has 1003 distinct LCF notations, with four of length two (illustrated above) and 999 of length 1. \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{Balaban graph : first form}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[scale=.6] \GraphInit[vstyle=Art] \SetGraphArtColor{red}{olive} \grBalaban[form=1,RA=7,RB=3,RC=3] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{Balaban graph : second form}} \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \GraphInit[vstyle=Art] \SetGraphArtColor{gray}{blue!50} \grBalaban[form=2,RA=7,RB=7,RC=4,RD=2.5] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{Balaban graph : third form} } \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture} \GraphInit[vstyle=Art] \SetGraphArtColor{brown}{orange} \grBalaban[form=3,RA=7,RB=6.5,RC=5.6,RD=5.6,RE=4.6] \end{tikzpicture} \end{tkzexample} \end{center} \vfill\newpage \subsection{\tkzname{Balaban graph : Balaban 11-Cage}} The Balaban 11-cage is the unique 11-cage graph, discovered by Balaban (1973) and proven unique by McKay and Myrvold (2003). It has 112 vertices, 168 edges, girth 11 (by definition), diameter 8 and chromatic number 3. \begin{center} \begin{tkzexample}[vbox] \begin{tikzpicture}[scale=.7] \renewcommand*{\VertexInnerSep}{3pt} \renewcommand*{\VertexLineWidth}{0.4pt} \GraphInit[vstyle=Art] \SetGraphArtColor{red!50}{blue!50!black} \grLCF[Math,RA=7]{% 44,26,-47,-15,35,-39,11,-27,38,-37,43,14,28,51,-29,-16,41,-11,% -26,15,22,-51,-35,36,52,-14,-33,-26,-46,52,26,16,43,33,-15,% 17,-53,23,-42,-35,-28,30,-22, 45,-44,16,-38,-16,50,-55,20,28,% -17,-43,47, 34,-26,-41,11,-36,-23,-16,41,17,-51,26,-33,47,17,% -11,-20 ,-30,21,29,36,-43,-52,10,39,-28,-17,-52,51,26,37,-17,% 10,-10,-45,-34,17,-26,27,-21,46,53,-10,29,-50,35,15,-47,-29,-41,% 26,33,55,-17,42,-26,-36,16}{1} \end{tikzpicture} \end{tkzexample} \end{center} \endinput