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\newpage\section{Cage}\label{cage}
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%<–––––––––––––––––––– Cage –––––––––––––––––––––––––––––––>
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\begin{NewMacroBox}{Cage Graphs}{}
\medskip
From Wikipedia \url{http://en.wikipedia.org/wiki/Cage_(graph_theory)}\\
\emph{In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth.\\
Formally, an $(r,g)$-graph is defined to be a graph in which each vertex has exactly $r$ neighbors, and in which the shortest cycle has length exactly $g$. It is known that an $(r,g)$-graph exists for any combination of $r \geq 2$ and $g \geq 3$. An $(r,g)$-cage is an $(r,g)$-graph with the fewest possible number of vertices, among all $(r,g)$-graphs.}
\medskip
From MathWorld \url{http://mathworld.wolfram.com/CageGraph.html}\\
\emph{A $(r,g)$-cage graph is a $v$-regular graph of girth $g$ having the minimum possible number of nodes. When $v$ is not explicitly stated, the term "$g$-cage" generally refers to a $(3,g)$-cage.}
\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
Examples :
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\begin{tabular}{ll}
\bottomrule
$(r,g)$ & Names \\
\midrule
$(3,3)$ & complete graph $K_4$ \\
$(3,4)$ & complete bipartite graph $K_{3,3}$ Utility Graph\ref{bipart} \\
$(3,5)$ & Petersen graph \ref{petersen} \\
$(3,6)$ & Heawood graph \ref{heawood} \\
$(3,7)$ & McGee graph \ref{mcgee} \\
$(3,8)$ & Levi graph \ref{levi} \\
$(3,10)$ & Balaban 10-cage \ref{balaban} \\
$(3,11)$ & Balaban 11-cage \ref{balaban} \\
$(3,12)$ & Tutte 12-cage \\
$(4,3)$ & complete graph $K_5$ \\
$(4,4)$ & complete bipartite graph $K_{4,4}$ \ref{bipart} \\
$(4,5)$ & Robertson graph\ref{robertson} \\
$(4,6)$ & Wong (1982)\ref{wong} \\
\end{tabular}
\end{NewMacroBox}
\endinput
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