% Copyright 2014, 2015, 2016 Kevin Ryde
%


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  % must be capital Latex for [harpoon] half arrows

\hyphenation{Amer-ican}
\hyphenation{Pos-it-ive}
\hyphenation{Calc-ul-at-ions}
\hyphenation{Rep-lic-at-ions}
\hyphenation{Twin-drag-on}
\hyphenation{Pap-er-fold-ing}
\hyphenation{Bit-Above-Low-est-One}
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\hyphenation{fract-al}
\hyphenation{trig-on-om-et-ric}
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%------------------------------------------------------------------------------
\begin{document}

binary beanstalk count 1-bits A179016
  parent = n - number of 1-bits in n

number of runs binary beanstalk A255327=size of subtree, A255056=trunk
  parent = A236840(vertex)   n - number of runs in n



%------------------------------------------------------------------------------
\section{Beineke All as Subgraphs, 12 Edges}

\url{https://hog.grinvin.org/ViewGraphInfo.action?id=748}

cf.\@ Introduction to Line Graphs
Emphasizing their construction, clique decompositions, and regularity

\begin{center}
\begin{tikzpicture}
    [scale=1,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]

  \node at (0,1) [my box,name=1] {1};
  \node at (0,2) [my box,name=3] {3};
  \node at (0,3) [my box,name=0] {0};
  \node at (0,4) [my box,name=2] {2};
  \node at (-2, .25) [my box,name=4] {4};
  \node at (2, .25) [my box,name=5] {5};

  \draw [] (0) to (4);
  \draw [] (0) to (2);
  \draw [] (2) to (5);
  \draw [] (1) to (4);
  \draw [] (0) to (5);
  \draw [] (3) to (5);
  \draw [] (4) to (5);
  \draw [] (0) to (3);
  \draw [] (3) to (1);
  \draw [] (3) to (4);
  \draw [] (2) to (4);
  \draw [] (1) to (5);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}
    [scale=2,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]

  \node at (4*72+18:1) [my box,name=0] {0};
  \node at (2*72+18:1) [my box,name=1] {1};
  \node at (0*72+18:1) [my box,name=2] {2};
  \node at (3*72+18:1) [my box,name=3] {3};
  \node at (0,0) [my box,name=4] {4};
  \node at (1*72+18:1) [my box,name=5] {5};

  \draw [] (0) to (4);
  \draw [] (0) to (2);
  \draw [] (2) to (5);
  \draw [] (1) to (4);
  \draw [] (0) to (5);
  \draw [] (3) to (5);
  \draw [] (4) to (5);
  \draw [] (0) to (3);
  \draw [] (3) to (1);
  \draw [] (3) to (4);
  \draw [] (2) to (4);
  \draw [] (1) to (5);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}
    [scale=2,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]

  \node at (0,0) [my box,name=0] {0};
  \node at (2,2) [my box,name=1] {1};
  \node at (2,0) [my box,name=2] {2};
  \node at (0,2) [my box,name=3] {3};
  \node at (1,1) [my box,name=4] {4};
  \node at (2,1) [my box,name=5] {5};

  \draw [] (0) to (4);
  \draw [] (0) to (2);
  \draw [] (2) to (5);
  \draw [] (1) to (4);
  \draw [] (0) to (5);
  \draw [] (3) to (5);
  \draw [] (4) to (5);
  \draw [] (0) to (3);
  \draw [] (3) to (1);
  \draw [] (3) to (4);
  \draw [] (2) to (4);
  \draw [] (1) to (5);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}
    [scale=2,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]

  \node at (0,0) [my box,name=0] {0};
  \node at (2,2) [my box,name=1] {1};
  \node at (2,0) [my box,name=2] {2};
  \node at (0,2) [my box,name=3] {3};
  \node at (.5,1) [my box,name=4] {4};
  \node at (1.5,1) [my box,name=5] {5};

  \draw [] (0) to (4);
  \draw [] (0) to (2);
  \draw [] (2) to (5);
  \draw [] (1) to (4);
  \draw [] (0) to (5);
  \draw [] (3) to (5);
  \draw [] (4) to (5);
  \draw [] (0) to (3);
  \draw [] (3) to (1);
  \draw [] (3) to (4);
  \draw [] (2) to (4);
  \draw [] (1) to (5);
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}
    [scale=2,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]

  \node at (1,0) [my box,name=1] {1};
  \node at (2,0) [my box,name=3] {3};
  \node at (3,0) [my box,name=0] {0};
  \node at (4,0) [my box,name=2] {2};
  \node at (2.5,1) [my box,name=4] {4};
  \node at (2.5,-1) [my box,name=5] {5};

  \draw [] (0) to (4);
  \draw [] (0) to (2);
  \draw [] (2) to (5);
  \draw [] (1) to (4);
  \draw [] (0) to (5);
  \draw [] (3) to (5);
  \draw [] (4) to (5);
  \draw [] (0) to (3);
  \draw [] (3) to (1);
  \draw [] (3) to (4);
  \draw [] (2) to (4);
  \draw [] (1) to (5);
\end{tikzpicture}
\end{center}




%------------------------------------------------------------------------------
\section{Beineke Subgraph Relations}

\begin{center}
\begin{tikzpicture}
    [scale=1.5,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]

  \node at (0,0) [my box,name=1] {1};
  \node at (-1,0) [my box,name=2] {2};
  \node at (-1,-1) [my box,name=3] {3};
  \node at (1,0) [my box,name=4] {4};
  \node at (-1,1) [my box,name=5] {5};
  \node at (0,1) [my box,name=6] {6};
  \node at (1, -1) [my box,name=7] {7};
  \node at (1,1) [my box,name=8] {8};
  \node at (0,-1) [my box,name=9] {9};

  \draw [->] (6) to (8);
  \draw [->] (6) to (1);
  \draw [->] (2) to (1);
  \draw [->] (8) to (4);
  \draw [->] (4) to (1);
  \draw [->,bend left=0] (5) to (4);
  \draw [->] (9) to (4);
  \draw [->] (6) to (4);
  \draw [->,bend right=0] (7) to (4);
  \draw [->] (5) to (2);
  \draw [->] (3) to (1);
  \draw [->] (5) to (1);
  \draw [->] (6) to (5);
  \draw [->] (9) to (1);
  \draw [->] (3) to (2);
  \draw [->] (8) to (1);
  \draw [->] (9) to (7);
  \draw [->] (6) to (7);
  \draw [->] (7) to (1);
  \draw [->,bend left=0] (6) to (2);

\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}
    [scale=2,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]

  \node at (0,0) [my box,name=1] {1};
  \node at (0*45+22.5:1) [my box,name=4] {4};
  \node at (1*45+22.5:1) [my box,name=8] {8};
  \node at (2*45+22.5:1) [my box,name=6] {6};
  \node at (3*45+22.5:1) [my box,name=5] {5};
  \node at (4*45+22.5:1) [my box,name=2] {2};
  \node at (5*45+22.5:1) [my box,name=3] {3};
  \node at (6*45+22.5:1) [my box,name=9] {9};
  \node at (7*45+22.5:1) [my box,name=7] {7};

  \draw [->] (6) to (8);
  \draw [->] (6) to (1);
  \draw [->] (2) to (1);
  \draw [->] (8) to (4);
  \draw [->] (4) to (1);
  \draw [->,bend left=0] (5) to (4);
  \draw [->] (9) to (4);
  \draw [->] (6) to (4);
  \draw [->,bend right=0] (7) to (4);
  \draw [->] (5) to (2);
  \draw [->] (3) to (1);
  \draw [->] (5) to (1);
  \draw [->] (6) to (5);
  \draw [->] (9) to (1);
  \draw [->] (3) to (2);
  \draw [->] (8) to (1);
  \draw [->] (9) to (7);
  \draw [->] (6) to (7);
  \draw [->] (7) to (1);
  \draw [->,bend left=0] (6) to (2);

\end{tikzpicture}
\end{center}


%------------------------------------------------------------------------------
\section{Hanoi 2 Discs, 4 Spindles, Star}

\url{https://hog.grinvin.org/ViewGraphInfo.action?id=21152}

\begin{center}
\begin{tikzpicture}
    [scale=1.25,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\scriptsize,
    ]
  % degree=2 count 9
  % degree=3 count 4
  % degree=4 count 3

  \node at (0,0)   [my box,name=00] {00};
  \node at (30:1)  [my box,name=01] {01};
  \node at (-90:1) [my box,name=02] {02};
  \node at (150:1) [my box,name=03] {03};

  \begin{scope}[shift={(210:2)}]
  \node at (0,0)   [my box,name=10] {10};
  \node at (210:1) [my box,name=11] {11};
  \node at (-30:1) [my box,name=12] {12};
  \node at (90:1) [my box,name=13] {13};
  \end{scope}

  \begin{scope}[shift={(90:2)}]
  \node at (0,0)   [my box,name=20] {20};
  \node at (-30:1) [my box,name=21] {21};
  \node at (90:1) [my box,name=22] {22};
  \node at (210:1) [my box,name=23] {23};
  \end{scope}

  \begin{scope}[shift={(-30:2)}]
  \node at (0,0)   [my box,name=30] {30};
  \node at (90:1) [my box,name=31] {31};
  \node at (210:1) [my box,name=32] {32};
  \node at (-30:1) [my box,name=33] {33};
  \end{scope}


  \draw [] (01) to (21);
  \draw [] (20) to (21);
  \draw [] (03) to (23);
  \draw [] (00) to (01);
  \draw [] (00) to (02);
  \draw [] (02) to (12);
  \draw [] (20) to (23);
  \draw [] (00) to (03);
  \draw [] (10) to (12);
  \draw [] (30) to (33);
  \draw [] (01) to (31);
  \draw [] (03) to (13);
  \draw [] (10) to (11);
  \draw [] (30) to (32);
  \draw [] (10) to (13);
  \draw [] (20) to (22);
  \draw [] (02) to (32);
  \draw [] (30) to (31);

\end{tikzpicture}
\end{center}

Each vertex is a configuration of discs on spindles for a variation by
Stockmeyer of the towers of Hanoi puzzle.  There are 4 spindles in a
star pattern and discs can only move between the centre spindle and
one of the 3 outer spindles (not among those outer spindles).

The present graph is for 2 discs.  The centre claw has big disc on the
centre spindle and the small disc moving to or from one of the outer
spindles.  When the small disc is not on the centre spindle the big
disc can move from there to one of the other outer spindles (where
further claws are the small disc moving).

The puzzle is to move both discs from one outer spindle to another outer 
spindle.  This is a path between two of the degree\hyp{}1 vertices. 

Paul K. Stockmeyer, "Variations on the Four-Post Tower of Hanoi
Puzzle", Congressus Numerantium, volume 102, 1994, pages 3-12,
\url{http://www.cs.wm.edu/~pkstoc/boca.ps}


%------------------------------------------------------------------------------
\section{Hanoi Graph 2 Discs, 4 Spindles, Linear}

\url{https://hog.grinvin.org/ViewGraphInfo.action?id=25143}

% 3, 10, 19, 34, 57, 88
% A160002
% my(n=2); 3^n+n-1 == 10
% my(n=3); 3^n+n-1 == 29

\begin{center}
\begin{tikzpicture}
  [scale=1.5,
   my box/.style={circle,draw,inner sep=.15em,outer sep=.3em,minimum size=1.6em},
  ]

  \node at (4,-1) [my box,name=0] {0};
  \node at (4,0) [my box,name=1] {1};
  \node at (4,1) [my box,name=2] {2};
  \node at (4,2) [my box,name=3] {3};

  \node at (1,1) [my box,name=4] {4};
  \node at (2,1) [my box,name=5] {5};
  \node at (3,1) [my box,name=6] {6};
  \node at (3,2) [my box,name=7] {7};

  \node at (0,1) [my box,name=8] {8};
  \node at (0,2) [my box,name=9] {9};
  \node at (1,2) [my box,name=10] {10};
  \node at (2,2) [my box,name=11] {11};

  \node at (-1,1) [my box,name=12] {12};
  \node at (-1,2) [my box,name=13] {13};
  \node at (-1,3) [my box,name=14] {14};
  \node at (-1,4) [my box,name=15] {15};


  \draw [] (14) to (15);
  \draw [] (9) to (10);
  \draw [] (0) to (1);
  \draw [] (9) to (8);
  \draw [] (4) to (5);
  \draw [] (3) to (7);
  \draw [] (1) to (2);
  \draw [] (7) to (11);
  \draw [] (6) to (7);
  \draw [] (4) to (8);
  \draw [] (12) to (8);
  \draw [] (9) to (13);
  \draw [] (2) to (6);
  \draw [] (13) to (14);
  \draw [] (2) to (3);
  \draw [] (6) to (5);
  \draw [] (12) to (13);
  \draw [] (10) to (11);

\end{tikzpicture}
\end{center}

Each vertex is a configuration of discs on spindles for a variation on the towers of Hanoi puzzle by Stockmeyer where discs can only move forward or backward between adjacent spindles along a row of 4 spindles.

The two degree-1 vertices are the 2 discs on the first or last
spindle.  The problem is to move the discs from one end to the other.
This is the graph diameter of 10 moves (OEIS A160002).

Paul K. Stockmeyer, "Variations on the Four-Post Tower of Hanoi
Puzzle", Congressus Numerantium, volume 102, 1994, pages 3-12,
\url{http://www.cs.wm.edu/~pkstoc/boca.ps}


% my(n=2); 3^n+n-1 == 10

Stockmeyer gives a straightforward recursive solution for the general case n discs along 4 spindles.  This establishes an upper bound of $3^n+n-1$ moves.  The n=2 case here is the last where that solution is optimal.

"The Four-in-a-Row Puzzle"

%------------------------------------------------------------------------------
\section{Hanoi Graph 2 Discs, 4 Spindles, Cyclic}

\url{https://hog.grinvin.org/ViewGraphInfo.action?id=25141}

\begin{center}
\begin{tikzpicture}
    [scale=2,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\scriptsize,
    ]

  \node at (0,0) [my box,name=0] {0};
  \node at (0,1) [my box,name=1] {1};
  \node at (1,1) [my box,name=2] {2};
  \node at (1,0) [my box,name=3] {3};

  \node at (3,1) [my box,name=4] {4};
  \node at (3,0) [my box,name=5] {5};
  \node at (2,0) [my box,name=6] {6};
  \node at (2,1) [my box,name=7] {7};

  \node at (2,2) [my box,name=8] {8};
  \node at (2,3) [my box,name=9] {9};
  \node at (3,3) [my box,name=10] {10};
  \node at (3,2) [my box,name=11] {11};

  \node at (1,3) [my box,name=12] {12};
  \node at (1,2) [my box,name=13] {13};
  \node at (0,2) [my box,name=14] {14};
  \node at (0,3) [my box,name=15] {15};

  \draw [] (2) to (6);
  \draw [] (1) to (13);
  \draw [] (13) to (14);
  \draw [] (7) to (4);
  \draw [] (1) to (2);
  \draw [] (15) to (14);
  \draw [] (11) to (10);
  \draw [] (4) to (5);
  \draw [] (8) to (4);
  \draw [] (6) to (7);
  \draw [] (12) to (13);
  \draw [] (8) to (12);
  \draw [] (0) to (3);
  \draw [] (9) to (10);
  \draw [] (8) to (9);
  \draw [] (0) to (1);
  \draw [] (8) to (11);
  \draw [] (3) to (7);
  \draw [] (6) to (5);
  \draw [] (3) to (2);
  \draw [] (12) to (15);
  \draw [] (2) to (14);
  \draw [] (9) to (13);
  \draw [] (11) to (7);

\end{tikzpicture}
\end{center}

Each vertex is a configuration of discs on spindles for a variation on the towers of Hanoi puzzle where discs can only move to an adjacent spindle around a cycle of 4 spindles.

The degree-2 vertices are where the 2 discs are on the same spindle so the only moves are the smaller disc back or forward.  The 4-cycle at each of those vertices is the small disc going successively around the 4 spindles.  The cross connections between those cycles are where the big disc moves (where permitted).

%------------------------------------------------------------------------------
\section{Hanoi 2 Discs 4 Spindles}

\url{https://hog.grinvin.org/ViewGraphInfo.action?id=22742}

\begin{center}
\begin{tikzpicture}
    [scale=.75,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\scriptsize,
    ]
  \newcommand\MyOuter{3}

  \begin{scope}[shift={(0,\MyOuter)}]
    \node at (0,1) [my box,name=0] {0};
    \node at (-1,0) [my box,name=1] {1};
    \node at (1,0) [my box,name=2] {2};
    \node at (0,-1) [my box,name=3] {3};
  \end{scope}

  \begin{scope}[shift={(\MyOuter,0)}]
    \node at (0,-1) [my box,name=4] {4};
    \node at (1,0) [my box,name=5] {5};
    \node at (-1,0) [my box,name=6] {6};
    \node at (0,1) [my box,name=7] {7};
  \end{scope}

  \begin{scope}[shift={(-\MyOuter,0)}]
    \node at (0,-1) [my box,name=8] {8};
    \node at (1,0) [my box,name=9] {9};
    \node at (-1,0) [my box,name=10] {10};
    \node at (0,1) [my box,name=11] {11};
  \end{scope}

  \begin{scope}[shift={(0,-\MyOuter)}]
    \node at (0,1) [my box,name=12] {12};
    \node at (-1,0) [my box,name=13] {13};
    \node at (1,0) [my box,name=14] {14};
    \node at (0,-1) [my box,name=15] {15};
  \end{scope}

  \draw [] (9) to (11);
  \draw [] (4) to (6);
  \draw [] (3) to (11);
  \draw [] (8) to (9);
  \draw [] (15) to (14);
  \draw [] (12) to (15);
  \draw [] (5) to (6);
  \draw [] (12) to (14);
  \draw [] (15) to (13);
  \draw [] (8) to (12);
  \draw [] (8) to (11);
  \draw [] (1) to (13);
  \draw [] (9) to (13);
  \draw [] (6) to (14);
  \draw [] (4) to (7);
  \draw [] (5) to (7);
  \draw [] (7) to (11);
  \draw [] (0) to (2);
  \draw [] (10) to (9);
  \draw [] (1) to (9);
  \draw [] (3) to (7);
  \draw [] (4) to (12);
  \draw [] (6) to (7);
  \draw [] (0) to (1);
  \draw [] (1) to (2);
  \draw [] (4) to (5);
  \draw [] (12) to (13);
  \draw [] (10) to (11);
  \draw [] (13) to (14);
  \draw [] (4) to (8);
  \draw [] (8) to (10);
  \draw [] (2) to (6);
  \draw [] (2) to (3);
  \draw [] (0) to (3);
  \draw [] (2) to (14);
  \draw [] (1) to (3);

\end{tikzpicture}
\end{center}

Each vertex is a configuration of discs on spindles for a variation on
the towers of Hanoi puzzle with 2 discs on 4 spindles.

The degree-3 vertices are where the 2 discs are on the same spindle so
only the smaller disc can move.  The complete-4 clique at each of
those is the small disc moving among the 4 spindles.  The cross
connections between those subgraphs are where the big disc moves (to
one of the 2 spindles different from itself and the smaller disc).



%------------------------------------------------------------------------------
\section{Johnson 5,2}

\url{https://hog.grinvin.org/ViewGraphInfo.action?id=21154}

\begin{center}
\begin{tikzpicture}
    [scale=.74,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]
  \newcommand\MyOuter{9}
  \newcommand\MyInner{6.2}
  % \newcommand\MyInnerAngle{54+180}
  \newcommand\MyInnerAngle{54}

  \node at (90-0*72: \MyOuter) [my box,name=1-2] {1,2};
  \node at (90-1*72: \MyOuter) [my box,name=2-3] {2,3};
  \node at (90-2*72: \MyOuter) [my box,name=3-4] {3,4};
  \node at (90-3*72: \MyOuter) [my box,name=4-5] {4,5};
  \node at (90-4*72: \MyOuter) [my box,name=1-5] {1,5};

  \node at (\MyInnerAngle-0*72: \MyInner) [my box,name=1-3] {1,3};
  \node at (\MyInnerAngle-1*72: \MyInner) [my box,name=2-4] {2,4};
  \node at (\MyInnerAngle-2*72: \MyInner) [my box,name=3-5] {3,5};
  \node at (\MyInnerAngle-3*72: \MyInner) [my box,name=1-4] {1,4};
  \node at (\MyInnerAngle-4*72: \MyInner) [my box,name=2-5] {2,5};

  \draw (1-2) to (1-3);
  \draw (1-2) to (1-4);
  \draw (1-2) to (1-5);
  \draw (1-2) to (2-3);
  \draw (1-2) to (2-4);
  \draw (1-2) to (2-5);
  
  \draw (1-4) to (4-5);
  \draw (2-4) to (3-4);
  \draw (1-3) to (1-5);
  \draw (1-5) to (2-5);
  \draw (1-3) to (2-3);
  \draw (2-5) to (4-5);
  \draw (3-5) to (4-5);
  \draw (1-3) to (3-4);
  \draw (2-3) to (3-5);
  \draw (3-4) to (4-5);
  \draw (1-4) to (1-5);
  \draw (1-5) to (3-5);
  \draw (1-4) to (3-4);
  \draw (1-4) to (2-4);
  \draw (1-3) to (1-4);
  \draw (2-3) to (3-4);
  \draw (3-4) to (3-5);
  \draw (2-3) to (2-4);
  \draw (1-3) to (3-5);
  \draw (2-5) to (3-5);
  \draw (2-4) to (4-5);
  \draw (1-5) to (4-5);
  \draw (2-4) to (2-5);
  \draw (2-3) to (2-5);

\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}
    [scale=1,
     my box/.style={circle,draw,inner sep=.15em,outer sep=.3em},
     font=\scriptsize,
    ]

  \node at (-0*36: 5) [my box,name=1-2] {1,2};    % 3
  \node at (-1*36: 5) [my box,name=2-5] {2,5};    % 7
  \node at (-2*36: 5) [my box,name=2-4] {2,4};    % 6
  \node at (-3*36: 5) [my box,name=2-3] {2,3};    % 5
  \node at (-4*36: 5) [my box,name=3-5] {3,5};    % 8
  \node at (-5*36: 5) [my box,name=3-4] {3,4};    % 7
  \node at (-6*36: 5) [my box,name=1-3] {1,3};    % 4

  \node at (-7*36: 5) [my box,name=1-5] {1,5};    % 6
  \node at (-8*36: 5) [my box,name=1-4] {1,4};    % 5
  \node at (-9*36: 5) [my box,name=4-5] {4,5};    % 9

  \draw (1-2) to (1-3);
  \draw (1-2) to (1-4);
  \draw (1-2) to (1-5);
  \draw (1-2) to (2-3);
  \draw (1-2) to (2-4);
  \draw (1-2) to (2-5);

  \draw (1-4) to (4-5);
  \draw (2-4) to (3-4);
  \draw (1-3) to (1-5);
  \draw (1-5) to (2-5);
  \draw (1-3) to (2-3);
  \draw (2-5) to (4-5);
  \draw (3-5) to (4-5);
  \draw (1-3) to (3-4);
  \draw (2-3) to (3-5);
  \draw (3-4) to (4-5);
  \draw (1-4) to (1-5);
  \draw (1-5) to (3-5);
  \draw (1-4) to (3-4);
  \draw (1-4) to (2-4);
  \draw (1-3) to (1-4);
  \draw (2-3) to (3-4);
  \draw (3-4) to (3-5);
  \draw (2-3) to (2-4);
  \draw (1-3) to (3-5);
  \draw (2-5) to (3-5);
  \draw (2-4) to (4-5);
  \draw (1-5) to (4-5);
  \draw (2-4) to (2-5);
  \draw (2-3) to (2-5);

\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}
    [scale=.74,
     my box/.style={circle,draw,minimum width=2.5em,inner sep=.15em,outer sep=.3em},
     font=\small,
    ]
  \newcommand\MyOuter{9}
  \newcommand\MyInner{5}
  \newcommand\MyInIn{5}
  % \newcommand\MyInnerAngle{54+180}
  \newcommand\MyInnerAngle{54}

  \node at (-0*36: \MyOuter) [my box,name=1-2] {1-2};
  \node at (-2*36: \MyOuter) [my box,name=2-3] {2-3};
  \node at (-4*36: \MyOuter) [my box,name=3-4] {3-4};
  \node at (-6*36: \MyOuter) [my box,name=4-5] {4-5};
  \node at (-8*36: \MyOuter) [my box,name=1-5] {1-5};

  \node at (-1*36: \MyInner) [my box,name=2-7] {2-7};
  \node at (-3*36: \MyInner) [my box,name=3-8] {3-8};
  \node at (-5*36: \MyInner) [my box,name=4-9] {4-9};
  \node at (-7*36: \MyInner) [my box,name=5-10] {5-10};
  \node at (-9*36: \MyInner) [my box,name=1-6] {1-6};

  \node at (-4*36: \MyInIn) [my box,name=10-7] {10-7};
  \node at (-6*36: \MyInIn) [my box,name=6-8] {6-8};
  \node at (-8*36: \MyInIn) [my box,name=7-9] {7-9};
  \node at (-10*36: \MyInIn) [my box,name=8-10] {8-10};
  \node at (-12*36: \MyInIn) [my box,name=6-9] {6-9};

  \draw [] (2-3) to (1-2);
  \draw [] (2-3) to (3-4);
  \draw [] (1-5) to (4-5);
  \draw [] (1-6) to (6-8);
  \draw [] (1-5) to (1-6);
  \draw [] (2-3) to (3-8);
  \draw [] (8-10) to (3-8);
  \draw [] (1-6) to (6-9);
  \draw [] (4-5) to (5-10);
  \draw [] (4-5) to (3-4);
  \draw [] (4-9) to (7-9);
  \draw [] (10-7) to (2-7);
  \draw [] (6-9) to (7-9);
  \draw [] (4-5) to (4-9);
  \draw [] (2-7) to (1-2);
  \draw [] (8-10) to (5-10);
  \draw [] (6-8) to (3-8);
  \draw [] (10-7) to (8-10);
  \draw [] (4-9) to (6-9);
  \draw [] (1-5) to (5-10);
  \draw [] (1-5) to (1-2);
  \draw [] (4-9) to (3-4);
  \draw [] (1-6) to (1-2);
  \draw [] (2-7) to (2-3);
  \draw [] (8-10) to (6-8);
  \draw [] (2-7) to (7-9);
  \draw [] (10-7) to (5-10);
  \draw [] (6-9) to (6-8);
  \draw [] (3-8) to (3-4);
  \draw [] (10-7) to (7-9);

\end{tikzpicture}
\end{center}

%------------------------------------------------------------------------------

\end{document}

% Local variables:
% compile-command: "latexmk -file-line-error -pdf misc.tex"
% End:
