fullgen is a program for generating nonisomorphic fullerenes.

Author: Gunnar Brinkmann (Gunnar.Brinkmann@Ugent.be).

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.Brinkmann@Ugent.be).

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

For 136 vertices or more (non-IPR), the following numbers are
different from earlier versions of this manual. In June 2011 Jan
Goedgebeur and Brendan McKay developed a new program (contact the
authors for a copy) for the generation of fullerenes that generated
different numbers for 136 vertices. It turned out that there was some
(non-algorithmic) error in fullgen that made the program miss
structures.  The smallest non-ipr case was 136 vertices where 1
structure was missed.  The smallest IPR case was at 254 vertices. This
error is now corrected.


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 |     5 566 949                        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 |     9 604 411                        192 |     9 364 975          
     146 |    10 867 631                        194 |    10 659 863          
     148 |    12 469 092                        196 |    12 163 298          
     150 |    14 059 174                        198 |    13 809 901          
     152 |    16 066 025                        200 |    15 655 672          
     154 |    18 060 979                        202 |    17 749 388          
     156 |    20 558 767                        204 |    20 070 486          
     158 |    23 037 594                        206 |    22 606 939          
     160 |    26 142 839                        208 |    25 536 557          
     162 |    29 202 543                        210 |    28 700 677          
     164 |    33 022 573                        212 |    32 230 861          
     166 |    36 798 433                        214 |    36 173 081          
     168 |    41 478 344                        216 |    40 536 922          
     170 |    46 088 157                        218 |    45 278 722          
     172 |    51 809 031                        220 |    50 651 799          
     174 |    57 417 264                        222 |    56 463 948          
     176 |    64 353 269                        224 |    62 887 775          
     178 |    71 163 452                        226 |    69 995 887          
     180 |    79 538 751                        228 |    77 831 323          
     182 |    87 738 311                        230 |    86 238 206          
     184 |    97 841 183                        232 |    95 758 929          
     186 |   107 679 717                        234 |   105 965 373          
     188 |   119 761 075                        236 |   117 166 528          
     190 |   131 561 744                        238 |   129 476 607          
     192 |   145 976 674                        240 |   142 960 479          
     194 |   159 999 462                        242 |   157 402 781
     196 |   177 175 687                        244 |   173 577 766
     198 |   193 814 658                        246 |   190 809 628
     200 |   214 127 742                        248 |   209 715 141
     202 |   233 846 463                        250 |   230 272 559
     204 |   257 815 889                        252 |   252 745 513
     206 |   281 006 325                        254 |   276 599 787
     208 |   309 273 526                        256 |   303 235 792
     210 |   336 500 830                        258 |   331 516 984
     212 |   369 580 714                        260 |   362 302 637
     214 |   401 535 955                        262 |   395 600 325
     216 |   440 216 206                        264 |   431 894 257
     218 |   477 420 176                        266 |   470 256 444
     220 |   522 599 564                        268 |   512 858 451
     222 |   565 900 181                        270 |   557 745 670
     224 |   618 309 598                        272 |   606 668 511
     226 |   668 662 698                        274 |   659 140 287
     228 |   729 414 880                        276 |   716 217 922
     230 |   787 556 069                        278 |   776 165 188
     232 |   857 934 016                        280 |   842 498 881
     234 |   925 042 498                        282 |   912 274 540
     236 | 1 006 016 526                        284 |   987 874 095
     238 | 1 083 451 816                        286 | 1 068 507 788
     240 | 1 176 632 247                        288 | 1 156 161 307
     242 | 1 265 323 971                        290 | 1 247 686 189
     244 | 1 372 440 782                        292 | 1 348 832 364
     246 | 1 474 111 053                        294 | 1 454 359 806
     248 | 1 596 482 232                        296 | 1 568 768 524
     250 | 1 712 934 069                        298 | 1 690 214 836
     252 | 1 852 762 875                        300 | 1 821 766 896
     254 | 1 985 250 572                
     256 | 2 144 943 655                
     258 | 2 295 793 276                
     260 | 2 477 017 558                
     262 | 2 648 697 036                
     264 | 2 854 536 850                
     266 | 3 048 609 900                
     268 | 3 282 202 941                
     270 | 3 501 931 260                
     272 | 3 765 465 341                
     274 | 4 014 007 928                
     276 | 4 311 652 376                
     278 | 4 591 045 471                
     280 | 4 926 987 377                
     282 | 5 241 548 270                
     284 | 5 618 445 787                
     286 | 5 972 426 835                
     288 | 6 395 981 131                
     290 | 6 791 769 082                
     292 | 7 267 283 603                
     294 | 7 710 782 991                
     296 | 8 241 719 706                
     298 | 8 738 236 515                
     300 | 9 332 065 811



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