fullgen is a program for generating nonisomorphic fullerenes.

Author: Gunnar Brinkmmann   gunnar@mathematik.uni-bielefeld.de

A fullerene is a 3-connected planar cubic graph whose faces are all
pentagons or hexagons.  From Euler's formula it can be shown that the
number of pentagons must be exactly 12.

An IPR fullerene is one for which no two pentagons share an edge.


The program fullgen must be called

fullgen x

with x the number of vertices of the fullerenes that shall be generated.

By default the fullerenes are just generated and counted. The results
are written to stderr and into a file named "Full_gen_x.log"

Several options can be given. Some of them have influence on the name of
the logfile (e.g. by adding a letter or two to the name) -- and all must 
be given after the number of vertices:

If you want not only fullerenes with a fixed number of vertices, but with 
vertex numbers in the range y to x, you can use the "start" option.
It is used e.g. "fullgen x start y" in this case. The time advantage between
generating the fullerenes with y, y+2, ... x vertices separately differs,
but is not very large. In the case of just generating ipr fullerenes the
advantage is larger.

Restricting the program to the generation of IPR fullerenes can be done by
using the option "ipr".

The generation can be split into 3 disjoint parts by using the options
"case 1", "case 2" or "case 3".

Another way to split the generation is to use "mod x y" with 0<=x<y. This way
is not as efficient as "case", but can be used additionally. One should not
use too large y. Different from "case" the splitting depends on ALL OPTIONS,
and the whole process, so if one first does "fullgen 60 start 20 mod 0 2" and
then "fullgen 60 mod 1 2" to get the rest of the 60 vertices fullerenes, the
result is WRONG. One would have to do "fullgen 60 start 20 mod 1 2".

You can choose whether you want the graphs not only to be counted, but
also coded. You can choose between 5 possibilities by typing e.g.
"fullgen x code z" with z the code you have chosen :

code 1: This is a binary code that is easy and fast to compute and to decode.
        Every entry of the code is an unsigned char. The first entry is the
        number of vertices.  After that there is a list of the vertices
        adjacent to vertex number 1 in clockwise orientation. This list is
        ended with a 0. Then you have the vertices adjacent to number 2 ended
        with a 0,... .  These codes are written into a file named
        "Full_codes_x".  In case that the numbers are too large for unsigned
        char, the first unsigned char written is 0 (!!!). After that the code
        is as above -- only with unsigned short instead of unsigned char.
        The unsigned shorts are written in little-endian order (low order
        byte first).  A 0 character is written before EVERY code.

code 2: This is a binary spiral code. Every entry is an unsigned char. For
        the complete spiral code you need the positions of the pentagons in
        a spiral development of the fullerene. So you need 12 entries giving
        the 12 positions. In fact the coding is a bit different: The first
        entry tells you how many entries you can take from the last code.
        So the first entry in every file is 0. The next 12 entries are code
        number 1. If the next entry is a 4, you take the first 4 entries of
        the first code and fill up the code using the next 8 entries from the
        file ...
        The files are called "Spiral_codes_x". The spiral developments
        chosen are completely arbitrary except that spirals starting at a
        pentagon are preferred.  In case a fullerene without spiral is found,
        it is written to "No_spiral_x" using code 1. Fullerenes without a
        spiral starting at a pentagon are written to "No_pentagon_spiral_x". 

code 3: This is a spiral code like code 2. The only difference is that in
        order to make the codes as equal as possible and obtain small files,
        the lexicographically minimal spiral development is calculated.
        Furthermore the codes are not written to the file one by one, but
        2000 codes are gathered and sorted in order to have similar codes
        close to each other. The number of graphs gathered can be changed by 
        using the option "list z" with z the number of codes that shall be
        gathered.  This is the most efficient and most time consuming code.

code 4: In this case the fullerenes are just counted, unless one without a
        spiral (or a spiral starting at a pentagon) occurs. This graph is
        treated as above.

code 5: This works completely like code 1 plus code 4 (concerning spirals)
        together.

code 6: This code is always written to stdout. The code used is the VEGA Code 
        >>writegraph3d planar<<

code 7: This code is also always written to stdout. It is the same code as
        in code 1, but the dual is written instead of the graph itself.

code 8: This is like code 1 but uses the sparse6 format (without header).
        See plantri-guide.txt for the definition of sparse6.

Except for code 6 and code 8, the output is preceded by the header
">>planar_code le<<" without end-of-line characters.

The option "spiralcheck" makes the program check for spirals independent of
the code used.

If the option "hexspi" is used, fullerenes that have no spirals starting at a
hexagon are looked for. If they are found, they are written to a file named 
"No_hexagon_spiral_x". It MUST be used in combination with some code involving
spiral checking or with the option spiralcheck.

The option "spistat" makes the program create some statistics about the
number of spirals the generated fullerenes have. The maximal possible number
is 4 times the number of edges, since for every edge one has two possibilities
to choose the initial face and for every such choice one has two possibilities
(clockwise and counterclockwise) for a spiral.  This does NOT mean that any
Non-spiral fullerenes are written unless you use other options too. This
option may be used only if fullerenes of ONE size are computed.

The option "symstat" makes the program create some statistics about the
symmetries which occur in the generated fullerenes. There are 28 possible
symmetry groups. 

If you want the graphs not only to be counted, but also coded, you can choose
if you want only graphs with a special symmetry to be coded. Use the option
"symm" by typing e.g.: "fullgen x code y symm z"
z must be one of the following 28 strings:
C1, C2, Ci, Cs, C3, D2, S4, C2v, C2h, D3, S6, C3v, C3h, D2h, D2d, D5, D6,
D3h, D3d, T, D5h, D5d, D6h, D6d, Td, Th, I, Ih.
These are standard names for various symmetry groups - please refer to a
chemical dictionary for their meanings.
You do not need to use the option "symstat" simultaneously.
Using the option "symm" doesn't decrease the generation time since all other
fullerenes are nevertheless generated, although they are not coded.
If you use the option "symm" but you don't use the option "code"
simultaneously, then the option "symm" has no effect. You can use the option
"symm" several times within one program call. Then you get every graph which
has one of the selected symmetries.
 
Please note that especially spistat takes quite some additional computing
time!  The option symstat takes about 2% additional computing time.

Output to files different from "No_spiral_x", "No_pentagon_spiral_x" and 
"No_hexagon_spiral_x" can be redirected to stdout by using the option stdout. 
This is useful e.g. for piping.

The option "quiet" makes fullgen suppress all information about the
generation process.

One possible call of fullgen would be

fullgen 150 start 100 ipr code 5 stdout | otherprogram

In case of problems or interesting results please contact 
Gunnar Brinkmann (gunnar@mathematik.uni-bielefeld.de).

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Numbers of Fullerenes and IPR Fullerenes:




                                         vertices | IPR-Fullerenes
                                         _________________________
                                               60 |           1 
                                               62 |           0 
vertices | Fullerenes                          64 |           0 
_______________________                        66 |           0 
      20 |           1                         68 |           0  
      22 |           0                         70 |           1 
      24 |           1                         72 |           1   
      26 |           1                         74 |           1  
      28 |           2                         76 |           2   
      30 |           3                         78 |           5  
      32 |           6                         80 |           7  
      34 |           6                         82 |           9  
      36 |          15                         84 |          24  
      38 |          17                         86 |          19   
      40 |          40                         88 |          35   
      42 |          45                         90 |          46   
      44 |          89                         92 |          86    
      46 |         116                         94 |         134     
      48 |         199                         96 |         187     
      50 |         271                         98 |         259    
      52 |         437                        100 |         450     
      54 |         580                        102 |         616     
      56 |         924                        104 |         823     
      58 |       1 205                        106 |       1 233     
      60 |       1 812                        108 |       1 799     
      62 |       2 385                        110 |       2 355    
      64 |       3 465                        112 |       3 342    
      66 |       4 478                        114 |       4 468   
      68 |       6 332                        116 |       6 063   
      70 |       8 149                        118 |       8 148  
      72 |      11 190                        120 |      10 774  
      74 |      14 246                        122 |      13 977  
      76 |      19 151                        124 |      18 769  
      78 |      24 109                        126 |      23 589  
      80 |      31 924                        128 |      30 683  
      82 |      39 718                        130 |      39 393  
      84 |      51 592                        132 |      49 878  
      86 |      63 761                        134 |      62 372  
      88 |      81 738                        136 |      79 362  
      90 |      99 918                        138 |      98 541  
      92 |     126 409                        140 |     121 354  
      94 |     153 493                        142 |     151 201  
      96 |     191 839                        144 |     186 611  
      98 |     231 017                        146 |     225 245  
     100 |     285 914                        148 |     277 930  
     102 |     341 658                        150 |     335 569  
     104 |     419 013                        152 |     404 667  
     106 |     497 529                        154 |     489 646  
     108 |     604 217                        156 |     586 264  
     110 |     713 319                        158 |     697 720  
     112 |     860 161                        160 |     836 497  
     114 |   1 008 444                        162 |     989 495  
     116 |   1 207 119                        164 |   1 170 157  
     118 |   1 408 553                        166 |   1 382 953  
     120 |   1 674 171                        168 |   1 628 029  
     122 |   1 942 929                        170 |   1 902 265  
     124 |   2 295 721                        172 |   2 234 133  
     126 |   2 650 866                        174 |   2 601 868  
     128 |   3 114 236                        176 |   3 024 383 
     130 |   3 580 637                        178 |   3 516 365 
     132 |   4 182 071                        180 |   4 071 832 
     134 |   4 787 715                        182 |   4 690 880 
     136 |   5 566 948                        184 |   5 424 777 
     138 |   6 344 698                        186 |   6 229 550 
     140 |   7 341 204                        188 |   7 144 091 
     142 |   8 339 033                        190 |   8 187 581 
     144 |   9 604 410                        192 |   9 364 975 
     146 |  10 867 629                        194 |  10 659 863 
     148 |  12 469 092                        196 |  12 163 298 
     150 |  14 059 173                        198 |  13 809 901 
     152 |  16 066 024                        200 |  15 655 672 
     154 |  18 060 973                        202 |  17 749 388 
     156 |  20 558 765                        204 |  20 070 486 
     158 |  23 037 593                        206 |  22 606 939 
     160 |  26 142 839                        208 |  25 536 557 
     162 |  29 202 540                        210 |  28 700 677 
     164 |  33 022 572                        212 |  32 230 861 
     166 |  36 798 430                        214 |  36 173 081 
     168 |  41 478 338                        216 |  40 536 922
     170 |  46 088 148                        218 |  45 278 722
     172 |  51 809 018                        220 |  50 651 799
     174 |  57 417 255                        222 |  56 463 948 
     176 |  64 353 257                        224 |  62 887 775
     178 |  71 163 435                        226 |  69 995 887
     180 |  79 538 725                        228 |  77 831 323
     182 |  87 738 289                        230 |  86 238 206 
     184 |  97 841 157                        232 |  95 758 929
     186 | 107 679 684                        234 | 105 965 373
     188 | 119 761 030                        236 | 117 166 528 
     190 | 131 561 725                        238 | 129 476 607
     192 | 145 976 654                        240 | 142 960 479 
     194 | 159 999 441                        
     196 | 177 175 662                        
     198 | 193 814 634                        
     200 | 214 127 713                        



Sample symmetry statistics for 60 vertices like outputted by Fullgen:

Symmetries:
  C1 :       1508   C2 :        189   Cs :         67   D2 :         19 
  S4 :          2   C2v:          9   C2h:          4   D3 :          3 
  C3v:          1   D2h:          1   D2d:          4   D5 :          1 
  D5d:          1   D6h:          2   Ih :          1