APPENDIX:
--------

The following are the several examples. The matrices are the adjacency
matrices of the graphs we consider.

 

[0 2 3 0
2 0 1 2
3 1 0 3
0 2 3 0 ]



      |V|       |E|    Density  Lowerbound Upperbound    Gap  
    4.0000    5.0000    1.2500    6.0000    6.0000         0  

      Upperbound Partition
     3     1

___________________________________________________________________________

W=[0 0 0 1
0 0 1 1
0 1 0 1
1 1 1 0 ]


      |V|       |E|    Density  Lowerbound Upperbound    Gap
       4         4          1     2             2         0

      Upperbound Partition
     3     2
-----------------------------------------------------------------------------
[0 2 3 0 1 2
2 0 3 0 1 2
3 3 0 3 0 1
0 0 3 0 2 3
1 1 0 2 0 0
2 2 1 3 0 0 ]


      |V|       |E|    Density  Lowerbound Upperbound    Gap  
    6.0000   11.0000    1.8333   10.0000   10.0000         0    

      Upperbound Partition
     1     2     3

---------------------------------------------------------------------------

W=[0 0 0 1 1 1
0 0 1 0 1 0
0 1 0 0 1 1
1 0 0 0 0 0
1 1 1 0 0 1
1 0 1 0 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
    6.0000    8.0000    1.3333    3.0000    3.0000         0

      Upperbound Partition
     5     3     2

----------------------------------------------------------------------------
[0 2 3 0 1 2 3 0
2 0 1 2 3 0 1 2
3 1 0 3 0 1 2 3
0 2 3 0 0 1 2 3
1 3 0 0 0 0 1 2
2 0 1 1 0 0 3 0
3 1 2 2 1 3 0 1
0 2 3 3 2 0 1 0 ]


      |V|       |E|    Density  Lowerbound Upperbound    Gap  
    8.0000   21.0000    2.6250   16.0000   16.0000         0

      Upperbound Partition
     3     1     7     6

----------------------------------------------------------------------------

W=[0 0 0 1 1 1 1 0
0 0 1 0 0 1 1 0
0 1 0 0 1 1 1 1
1 0 0 0 1 0 0 0
1 0 1 1 0 1 1 1
1 1 1 0 1 0 0 1
1 1 1 0 1 0 0 1
0 0 1 0 1 1 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
    8.0000   17.0000    2.1250    7.0000    7.0000         0

      Upperbound Partition
     7     5     1     4

-----------------------------------------------------------------------------

[0 2 3 0 1 2 3 0 1 2
2 0 3 0 1 2 3 0 1 2
3 3 0 3 0 1 2 3 0 1
0 0 3 0 2 3 0 1 2 3
1 1 0 2 0 0 1 2 3 0
2 2 1 3 0 0 1 2 3 0
3 3 2 0 1 1 0 1 2 3
0 0 3 1 2 2 1 0 0 1
1 1 0 2 3 3 2 0 0 2
2 2 1 3 0 0 3 1 2 0 ]


      |V|       |E|    Density  Lowerbound Upperbound    Gap  
   10.0000   34.0000    3.4000   26.0000   26.0000         0 

      Upperbound Partition
     8     9     5     4     6

----------------------------------------------------------------------------
W=[0 0 0 1 1 1 1 0 1 0
0 0 0 1 1 0 0 1 1 1
0 0 0 1 1 0 0 0 1 1
1 1 1 0 1 0 1 1 0 0
1 1 1 1 0 0 1 0 0 1
1 0 0 0 0 0 1 0 1 1
1 0 0 1 1 1 0 0 0 0
0 1 0 1 0 0 0 0 1 1
1 1 1 0 0 1 0 1 0 1
0 1 1 0 1 1 0 1 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   10.0000   25.0000    2.5000    9.0000    9.0000         0

      Upperbound Partition
     6     2     9     8    10

-----------------------------------------------------------------------------

W=[0 2 2 1 3 3 3 2 3 0 2 1
2 0 3 0 0 1 3 1 3 1 2 2
2 3 0 2 3 3 3 2 3 1 2 0
1 0 2 0 0 1 2 2 3 3 2 1
3 0 3 0 0 1 0 0 0 1 3 3
3 1 3 1 1 0 2 1 3 2 2 1
3 3 3 2 0 2 0 0 0 3 2 2
2 1 2 2 0 1 0 0 2 3 1 1
3 3 3 3 0 3 0 2 0 3 3 3
0 1 1 3 1 2 3 3 3 0 2 2
2 2 2 2 3 2 2 1 3 2 0 3
1 2 0 1 3 1 2 1 3 2 3 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap  
   12.0000   56.0000    4.6667   53.0000   54.0000    0.0189 

      Upperbound Partition
     5     6     3     2     1     7

-----------------------------------------------------------------------------
W=[0 0 0 1 1 1 1 0 1 0 0 1
0 0 1 0 0 1 1 1 1 1 0 0
0 1 0 0 1 1 1 0 1 1 0 0
1 0 0 0 0 1 0 0 1 1 0 1
1 0 1 0 0 1 0 0 0 1 1 1
1 1 1 1 1 0 0 1 1 0 0 1
1 1 1 0 0 0 0 0 0 1 0 0
0 1 0 0 0 1 0 0 0 1 1 1
1 1 1 1 0 1 0 0 0 1 1 1
0 1 1 1 1 0 1 1 1 0 0 0
0 0 0 0 1 0 0 1 1 0 0 1
1 0 0 1 1 1 0 1 1 0 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   12.0000   36.0000    3.0000   13.0000   14.0000    0.0769

      Upperbound Partition
     5     9     4    12     6     1

------------------------------------------------------------------------------

W=[0 2 2 1 3 3 3 2 3 0 2 1
2 0 3 0 0 1 3 1 3 1 2 2
2 3 0 2 3 3 3 2 3 1 2 0
1 0 2 0 0 1 2 2 3 3 2 1
3 0 3 0 0 1 0 0 0 1 3 3
3 1 3 1 1 0 2 1 3 2 2 1
3 3 3 2 0 2 0 0 0 3 2 2
2 1 2 2 0 1 0 0 2 3 1 1
3 3 3 3 0 3 0 2 0 3 3 3
0 1 1 3 1 2 3 3 3 0 2 2
2 2 2 2 3 2 2 1 3 2 0 3
1 2 0 1 3 1 2 1 3 2 3 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   12.0000   56.0000    4.6667   53.0000   54.0000    0.0189
      Upperbound Partition
     5     6     3     2     1     7

-------------------------------------------------------------------------------
W=[0 0 0 1 1 1 1 0 1 0 0 1 1 0
0 0 0 1 1 1 1 1 0 0 0 1 1 1
0 0 0 0 1 1 0 0 0 1 0 0 1 1
1 1 0 0 0 1 1 0 0 0 1 1 1 0
1 1 1 0 0 1 1 0 0 1 0 0 1 0
1 1 1 1 1 0 0 0 1 1 1 1 1 1
1 1 0 1 1 0 0 0 0 1 0 1 1 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 1 0 0 0 0 1 1 0 1
0 0 1 0 1 1 1 0 0 0 0 1 0 0
0 0 0 1 0 1 0 0 1 0 0 1 1 1
1 1 0 1 0 1 1 0 1 1 1 0 0 1
1 1 1 1 1 1 1 0 0 0 1 0 0 0
0 1 1 0 0 1 0 0 1 0 1 1 0 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   14.0000   46.0000    3.2857   17.0000   19.0000    0.1176

      Upperbound Partition
     3     1    14    12     6    11     9
------------------------------------------------------------------------------
W=[0 2 2 1 3 3 3 2 3 0 2 1 3 0
2 0 0 1 3 1 3 1 2 2 2 3 3 3
2 0 0 2 3 1 2 0 0 1 2 2 3 3
1 1 2 0 2 1 1 0 0 0 1 3 3 2
3 3 3 2 0 1 3 2 2 1 0 0 3 2
3 1 1 1 1 0 2 2 3 1 1 3 3 3
3 3 2 1 3 2 0 2 2 3 0 1 1 2
2 1 0 0 2 2 2 0 2 2 2 0 2 2
3 2 0 0 2 3 2 2 0 0 1 1 2 3
0 2 1 0 1 1 3 2 0 0 2 1 0 0
2 2 2 1 0 1 0 2 1 2 0 3 3 1
1 3 2 3 0 3 1 0 1 1 3 0 0 1
3 3 3 3 3 3 1 2 2 0 3 0 0 0
0 3 3 2 2 3 2 2 3 0 1 1 0 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   14.0000   74.0000    5.2857   66.0000   69.0000    0.0455

      Upperbound Partition
    10     6    13    12     3    11     4

------------------------------------------------------------------------------
W=[0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1
0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0
0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 0
1 1 0 0 1 1 1 0 1 1 0 0 1 0 0 1
1 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0
1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0
1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0
0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1
1 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0
0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0
0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0
1 1 1 0 1 0 0 1 1 0 1 0 1 0 1 0
1 0 1 1 1 0 1 0 1 1 0 1 0 1 0 0
0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0
0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 1
1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   16.0000   60.0000    3.7500   22.0000   22.0000         0

      Upperbound Partition
     3    10    13     1     7     9     4     6

-------------------------------------------------------------------------------

W=[0 2 2 1 3 3 3 2 3 0 2 1 3 0 0 1
2 0 3 1 3 1 2 2 2 3 3 3 2 3 1 2
2 3 0 0 0 1 2 2 3 3 2 1 1 0 0 0
1 1 0 0 1 3 3 2 1 3 2 2 1 0 0 3
3 3 0 1 0 2 2 2 3 1 1 3 3 3 2 2
3 1 1 3 2 0 3 0 1 1 2 2 2 2 0 2
3 2 2 3 2 3 0 2 0 1 1 2 3 2 1 0
2 2 2 2 2 0 2 0 0 3 3 1 0 1 0 1
3 2 3 1 3 1 0 0 0 1 0 3 1 2 3 0
0 3 3 3 1 1 1 3 1 0 1 2 3 2 1 2
2 3 2 2 1 2 1 3 0 1 0 3 2 1 2 2
1 3 1 2 3 2 2 1 3 2 3 0 3 0 1 2
3 2 1 1 3 2 3 0 1 3 2 3 0 1 2 0
0 3 0 0 3 2 2 1 2 2 1 0 1 0 2 0
0 1 0 0 2 0 1 0 3 1 2 1 2 2 0 1
1 2 0 3 2 2 0 1 0 2 2 2 0 0 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   16.0000   98.0000    6.1250   84.0000   84.0000         0
      Upperbound Partition
    12     2     4     3    16    10     8    11
-------------------------------------------------------------------------------
W=[0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 1 1
0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 0
0 1 0 0 1 1 0 1 1 0 0 0 1 1 1 0 1 1
1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1
1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0
1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 0 1 1
1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 0 1
0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1
1 1 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0
0 1 0 1 0 1 1 1 0 0 0 0 1 0 0 0 1 1
0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0
1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 1 0 0
1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1
0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1
1 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 1
1 1 1 1 0 1 0 0 0 1 1 0 1 1 1 1 0 1
1 0 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   18.0000   79.0000    4.3889   29.0000   29.0000         0

      Upperbound Partition
    11     8    14     2    12     3     5     9     6
---------------------------------------------------------------------------

W=[0 2 2 1 3 3 3 2 3 0 2 1 3 0 0 1 3 1
2 0 3 1 2 2 2 3 3 3 2 3 1 2 0 0 1 2
2 3 0 2 3 3 2 1 1 0 0 0 1 3 3 2 1 3
1 1 2 0 2 2 1 0 0 3 2 2 2 3 1 1 3 3
3 2 3 2 0 3 2 2 3 0 1 1 2 2 2 2 0 2
3 2 3 2 3 0 2 0 1 1 2 3 2 1 0 0 3 3
3 2 2 1 2 2 0 1 0 1 0 1 1 0 3 1 2 3
2 3 1 0 2 0 1 0 0 1 2 3 2 1 2 3 2 1
3 3 1 0 3 1 0 0 0 2 2 3 0 1 2 1 2 0
0 3 0 3 0 1 1 1 2 0 2 0 1 0 2 2 1 1
2 2 0 2 1 2 0 2 2 2 0 2 3 3 2 1 1 2
1 3 0 2 1 3 1 3 3 0 2 0 2 0 0 1 0 0
3 1 1 2 2 2 1 2 0 1 3 2 0 3 0 1 3 1
0 2 3 3 2 1 0 1 1 0 3 0 3 0 1 2 3 0
0 0 3 1 2 0 3 2 2 2 2 0 0 1 0 1 3 1
1 0 2 1 2 0 1 3 1 2 1 1 1 2 1 0 3 3
3 1 1 3 0 3 2 2 2 1 1 0 3 3 3 3 0 3
1 2 3 3 2 3 3 1 0 1 2 0 1 0 1 3 3 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   18.0000  124.0000    6.8889  103.0000  106.0000    0.0291

      Upperbound Partition
    13    11     8     6     5     1    12     9     2

----------------------------------------------------------------------------
[0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 ];

      |V|       |E|    Density  Lowerbound Upperbound    Gap  
   20.0000   95.0000    4.7500   26.0000   26.0000         0 

      Upperbound Partition
    11    15    19    17    13     9     5     2     1     6

---------------------------------------------------------------------------

W=[0 2 2 1 3 3 3 2 3 0 2 1 3 0 0 1 3 1 3 1
2 0 2 2 2 3 3 3 2 3 1 2 0 0 1 2 2 3 3 2
2 2 0 1 1 0 0 0 1 3 3 2 1 3 2 2 1 0 0 3
1 2 1 0 2 2 2 3 1 1 3 3 3 2 2 3 0 1 1 2
3 2 1 2 0 2 2 2 0 2 2 0 1 1 2 3 2 1 0 0
3 3 0 2 2 0 3 3 1 0 1 0 1 1 0 3 1 2 3 0
3 3 0 2 2 3 0 1 2 3 2 1 2 3 2 1 2 2 3 0
2 3 0 3 2 3 1 0 1 2 1 2 0 2 0 1 0 2 2 1
3 2 1 1 0 1 2 1 0 1 2 3 3 2 1 1 2 2 0 0
0 3 3 1 2 0 3 2 1 0 1 0 0 3 0 1 3 1 1 2
2 1 3 3 2 1 2 1 2 1 0 3 0 1 3 1 3 3 3 1
1 2 2 3 0 0 1 2 3 0 3 0 1 0 1 1 1 1 3 3
3 0 1 3 1 1 2 0 3 0 0 1 0 1 2 2 3 2 2 0
0 0 3 2 1 1 3 2 2 3 1 0 1 0 0 2 2 1 3 3
0 1 2 2 2 0 2 0 1 0 3 1 2 0 0 1 0 0 3 3
1 2 2 3 3 3 1 1 1 1 1 1 2 2 1 0 2 0 0 1
3 2 1 0 2 1 2 0 2 3 3 1 3 2 0 2 0 3 3 1
1 3 0 1 1 2 2 2 2 1 3 1 2 1 0 0 3 0 0 3
3 3 0 1 0 3 3 2 0 1 3 3 2 3 3 0 3 0 0 2
1 2 3 2 0 0 0 1 0 2 1 3 0 3 3 1 1 3 2 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   20.0000  153.0000    7.6500  127.0000  134.0000    0.0551

      Upperbound Partition
    11     2    19     6    13     9     7    18    17     1
------------------------------------------------------------------------------
W=[0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   30.0000  217.0000    7.2333   56.0000   56.0000         0

      Upperbound Partition
 14  13  9  5  1  2   6  10  19  23  27  29 25  21 17
---------------------------------------------------------------------------


W=[0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap  
   1.0e+04 *
    0.0040    0.0390    0.0010    0.0100    0.0100         0 

      Upperbound Partition
18  23  14  27  10  31   6  35  39  2  1  37 5  33  9  29 13  25  17 21
----------------------------------------------------------------------------

 
W=[0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 1 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 ]

       |V|       |E|    Density  Lowerbound Upperbound    Gap
   50.0000  612.0000   12.2400  156.0000  156.0000         0

      Upperbound Partition
   27    31    35    39    43    47    49    45    41    37    33    29
25    22    18    14    10     6     2     1     5     9    13    17    21
-----------------------------------------------------------------------
W=[0 2 2 1 3 3 3 2 3 0 2 1 3 0 0 1 3 1 3 1 2 2 2 3 3 3 2 3 1 2 0 0 1 2 2 3 3 2 1 1 0 0 0 1 3 3
2 0 2 1 3 2 2 1 0 0 3 2 2 2 3 1 1 3 3 3 2 2 3 0 1 1 2 2 2 2 0 2 2 0 1 1 2 3 2 1 0 0 3 3 1 0
2 2 0 1 0 1 1 0 3 1 2 3 0 1 2 3 2 1 2 3 2 1 2 2 3 0 1 2 1 2 0 2 0 1 0 2 2 1 1 2 3 3 2 1 1 2
1 1 1 0 2 0 0 1 0 0 3 0 1 3 1 1 2 3 0 1 3 1 3 3 3 1 1 0 1 1 1 1 3 3 1 2 2 3 2 2 0 0 2 2 1 3
3 3 0 2 0 3 1 0 0 3 3 2 0 0 1 3 3 1 0 3 2 3 1 0 1 1 3 1 2 1 0 3 2 2 1 3 0 3 0 1 1 2 2 3 2 0
3 2 1 0 3 0 0 2 3 3 0 2 1 2 0 0 3 0 3 3 1 0 2 3 2 0 0 1 2 3 1 3 3 0 0 0 3 3 0 2 2 3 0 3 1 1
3 2 1 0 1 0 0 2 2 2 3 0 1 3 1 2 2 1 0 3 1 1 3 2 2 1 2 3 0 0 1 2 1 3 1 3 0 2 1 3 3 2 2 0 1 1
2 1 0 1 0 2 2 0 1 2 3 3 2 1 2 3 1 2 0 0 0 1 3 0 1 0 3 3 0 3 3 1 2 3 0 2 1 1 1 1 0 0 3 3 1 0
3 0 3 0 0 3 2 1 0 1 1 3 0 3 3 1 0 1 0 0 3 2 1 2 3 3 1 3 2 2 2 3 0 0 2 1 2 0 1 2 2 0 1 2 1 1
0 0 1 0 3 3 2 2 1 0 3 2 1 3 3 2 2 0 1 0 2 0 3 1 0 0 3 1 2 2 2 2 2 0 1 0 1 1 2 1 2 0 3 3 0 3
2 3 2 3 3 0 3 3 1 3 0 1 3 3 3 0 1 1 2 2 0 1 2 0 3 1 3 1 0 2 1 1 1 3 2 1 1 1 3 3 3 1 3 3 2 1
1 2 3 0 2 2 0 3 3 2 1 0 1 1 3 1 1 3 0 1 0 2 0 2 3 0 2 3 1 2 2 1 1 2 2 3 3 0 3 0 2 1 3 0 1 1
3 2 0 1 0 1 1 2 0 1 3 1 0 3 0 2 1 0 0 2 0 1 1 3 2 2 1 0 2 2 0 1 3 0 0 2 1 3 1 1 1 0 3 0 3 0
0 2 1 3 0 2 3 1 3 3 3 1 3 0 3 0 0 1 1 1 3 1 2 2 3 1 1 0 0 3 2 3 2 2 1 2 3 3 0 2 2 3 0 0 0 3
0 3 2 1 1 0 1 2 3 3 3 3 0 3 0 2 2 1 2 2 0 2 2 1 0 0 2 1 3 1 2 0 1 3 1 0 2 1 2 1 3 1 1 2 3 3
1 1 3 1 3 0 2 3 1 2 0 1 2 0 2 0 0 1 3 3 1 1 3 1 2 1 1 1 2 2 3 1 3 2 1 1 0 2 2 0 1 0 2 1 1 0
3 1 2 2 3 3 2 1 0 2 1 1 1 0 2 0 0 0 1 3 0 0 3 3 0 2 1 2 0 1 0 1 0 2 2 2 0 2 1 1 3 2 3 3 1 1
1 3 1 3 1 0 1 2 1 0 1 3 0 1 1 1 0 0 3 1 3 3 1 3 3 2 0 3 1 1 1 1 1 1 0 3 2 0 2 2 2 0 1 2 2 3
3 3 2 0 0 3 0 0 0 1 2 0 0 1 2 3 1 3 0 2 2 3 3 1 2 0 0 3 1 2 2 3 3 1 1 2 0 3 0 2 3 2 3 1 3 0
1 3 3 1 3 3 3 0 0 0 2 1 2 1 2 3 3 1 2 0 3 0 3 1 3 2 3 0 1 3 3 1 3 0 0 1 3 3 0 1 1 3 3 0 2 1
2 2 2 3 2 1 1 0 3 2 0 0 0 3 0 1 0 3 2 3 0 2 3 0 2 3 2 0 0 2 3 0 3 1 3 1 0 2 3 1 3 2 0 1 0 2
2 2 1 1 3 0 1 1 2 0 1 2 1 1 2 1 0 3 3 0 2 0 3 1 1 1 2 3 2 0 1 0 2 0 3 2 1 1 0 1 0 2 3 1 0 0
2 3 2 3 1 2 3 3 1 3 2 0 1 2 2 3 3 1 3 3 3 3 0 2 1 1 1 3 0 3 1 0 1 3 3 3 3 3 0 3 2 0 3 1 0 3
3 0 2 3 0 3 2 0 2 1 0 2 3 2 1 1 3 3 1 1 0 1 2 0 0 3 2 1 3 1 0 0 1 2 3 1 1 1 1 0 0 0 1 1 1 1
3 1 3 3 1 2 2 1 3 0 3 3 2 3 0 2 0 3 2 3 2 1 1 0 0 3 0 3 3 2 0 3 3 0 0 1 2 0 3 0 2 2 3 0 0 3
3 1 0 1 1 0 1 0 3 0 1 0 2 1 0 1 2 2 0 2 3 1 1 3 3 0 3 0 0 2 2 3 3 1 0 3 2 1 0 0 3 0 3 2 0 2
2 2 1 1 3 0 2 3 1 3 3 2 1 1 2 1 1 0 0 3 2 2 1 2 0 3 0 1 2 3 1 1 0 0 3 1 2 1 1 0 0 2 1 1 0 1
3 2 2 0 1 1 3 3 3 1 1 3 0 0 1 1 2 3 3 0 0 3 3 1 3 0 1 0 2 0 0 0 3 3 1 1 1 0 1 2 0 0 0 0 3 3
1 2 1 1 2 2 0 0 2 2 0 1 2 0 3 2 0 1 1 1 0 2 0 3 3 0 2 2 0 1 0 0 0 3 0 2 0 2 1 1 2 1 2 3 3 1
2 2 2 1 1 3 0 3 2 2 2 2 2 3 1 2 1 1 2 3 2 0 3 1 2 2 3 0 1 0 0 0 2 0 3 3 0 3 3 0 0 2 3 1 2 0
0 0 0 1 0 1 1 3 2 2 1 2 0 2 2 3 0 1 2 3 3 1 1 0 0 2 1 0 0 0 0 2 3 1 2 0 1 2 1 3 3 0 3 2 0 0
0 2 2 1 3 3 2 1 3 2 1 1 1 3 0 1 1 1 3 1 0 0 0 0 3 3 1 0 0 0 2 0 2 2 0 0 2 2 0 2 2 0 1 2 1 1
1 2 0 3 2 3 1 2 0 2 1 1 3 2 1 3 0 1 3 3 3 2 1 1 3 3 0 3 0 2 3 2 0 3 2 0 2 2 2 1 1 2 2 2 2 1
2 0 1 3 2 0 3 3 0 0 3 2 0 2 3 2 2 1 1 0 1 0 3 2 0 1 0 3 3 0 1 2 3 0 3 1 1 1 0 2 2 0 2 2 3 3
2 1 0 1 1 0 1 0 2 1 2 2 0 1 1 1 2 0 1 0 3 3 3 3 0 0 3 1 0 3 2 0 2 3 0 2 3 2 3 2 1 3 1 2 3 1
3 1 2 2 3 0 3 2 1 0 1 3 2 2 0 1 2 3 2 1 1 2 3 1 1 3 1 1 2 3 0 0 0 1 2 0 0 2 1 3 1 1 1 0 0 0
3 2 2 2 0 3 0 1 2 1 1 3 1 3 2 0 0 2 0 3 0 1 3 1 2 2 2 1 0 0 1 2 2 1 3 0 0 0 2 3 1 2 0 2 3 1
2 3 1 3 3 3 2 1 0 1 1 0 3 3 1 2 2 0 3 3 2 1 3 1 0 1 1 0 2 3 2 2 2 1 2 2 0 0 0 1 1 2 3 2 2 2
1 2 1 2 0 0 1 1 1 2 3 3 1 0 2 2 1 2 0 0 3 0 0 1 3 0 1 1 1 3 1 0 2 0 3 1 2 0 0 3 0 2 2 1 1 2
1 1 2 2 1 2 3 1 2 1 3 0 1 2 1 0 1 2 2 1 1 1 3 0 0 0 0 2 1 0 3 2 1 2 2 3 3 1 3 0 1 1 2 2 1 2
0 0 3 0 1 2 3 0 2 2 3 2 1 2 3 1 3 2 3 1 3 0 2 0 2 3 0 0 2 0 3 2 1 2 1 1 1 1 0 1 0 1 3 0 3 0
0 0 3 0 2 3 2 0 0 0 1 1 0 3 1 0 2 0 2 3 2 2 0 0 2 0 2 0 1 2 0 0 2 0 3 1 2 2 2 1 1 0 3 1 0 0
0 3 2 2 2 0 2 3 1 3 3 3 3 0 1 2 3 1 3 3 0 3 3 1 3 3 1 0 2 3 3 1 2 2 1 1 0 3 2 2 3 3 0 3 3 2
1 3 1 2 3 3 0 3 2 3 3 0 0 0 2 1 3 2 1 0 1 1 1 1 0 2 1 0 3 1 2 2 2 2 2 0 2 2 1 2 0 1 3 0 0 3
3 1 1 1 2 1 1 1 1 0 2 1 3 0 3 1 1 2 3 2 0 0 0 1 0 0 0 3 3 2 0 1 2 3 3 0 3 2 1 1 3 0 3 0 0 2
3 0 2 3 0 1 1 0 1 3 1 1 0 3 3 0 1 3 0 1 2 0 3 1 3 2 1 3 1 0 0 1 1 3 1 0 1 2 2 2 0 0 2 3 2 0 ]

      |V|       |E|    Density  Lowerbound Upperbound    Gap
   46.0000  795.0000   17.2826  668.0000  687.0000    0.0284

      Upperbound Partition
  29    23    14    25     7    45    15     3    36    22     1    34
     24     9    12    40     4    35    37    28    18    46    39

