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author | Norbert Preining <norbert@preining.info> | 2021-01-22 03:00:46 +0000 |
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committer | Norbert Preining <norbert@preining.info> | 2021-01-22 03:00:46 +0000 |
commit | e1a7798f45cd37c0aa0bfaaef5f0acb4e48bc867 (patch) | |
tree | d2bbebcdb7bcb8fe013a9d3cbd2cff880b18afbf /macros/latex/contrib/tkz/tkz-berge/NamedGraphs/doc/latex/NamedGraphs-Cage.tex | |
parent | 37cb9bbabbfe1a8d0d1334832814ccbefcf9ac0a (diff) |
CTAN sync 202101220300
Diffstat (limited to 'macros/latex/contrib/tkz/tkz-berge/NamedGraphs/doc/latex/NamedGraphs-Cage.tex')
-rw-r--r-- | macros/latex/contrib/tkz/tkz-berge/NamedGraphs/doc/latex/NamedGraphs-Cage.tex | 43 |
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diff --git a/macros/latex/contrib/tkz/tkz-berge/NamedGraphs/doc/latex/NamedGraphs-Cage.tex b/macros/latex/contrib/tkz/tkz-berge/NamedGraphs/doc/latex/NamedGraphs-Cage.tex new file mode 100644 index 0000000000..520b38555f --- /dev/null +++ b/macros/latex/contrib/tkz/tkz-berge/NamedGraphs/doc/latex/NamedGraphs-Cage.tex @@ -0,0 +1,43 @@ +\newpage\section{Cage}\label{cage} +%<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> +%<–––––––––––––––––––– Cage –––––––––––––––––––––––––––––––> +%<––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> +\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 : + +\medskip +\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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