All graphs are stored in the Graph6-format, see Brendan Mackay's
webpage for documentation on this format. The graphs were constructed
by an exahustive search for each number of vertices using a modified
version of Gunnar Brinkmann's generator for cubic graphs minibaum.
For small N the search was laso done by using an unmodified version of
minibaum together with a Fortran90-program which filtered out graphs
with unwanted cycles.
In a file named cubic_noX_Y_Z_..._nN.g6
you will find all 3-connected cubic graphs on N vertices which do not
contain cycles of length X, Y, Z...
I have done a complete search for these graphs up to the largest N
given here for each given combination of cycle lengths. So if there is
no file for a given N this means that there are no 3-connected cubic
graphs on that many vertices avoiding the given cycles.
I have looked for cubic graphs with no cycles of lengths
4,8,16. If have found no such graphs and have searched all N<=52
This was in part done as a test of a conjecture of Erdös and
Gyarfas. See this paper for more
information. Geoff Exoo has found a graph of this kind with 78
vertices.
I have done a search for the smallest graphs with no
cycles of lengths 4,6,8,10,12. I have searched up to N<=66 and found no
such graphs.
Geoff Exoo has also found several
other graphs avoiding certain cycles length, some of which are now
known to be the smallest of their kind by this exhaustive search.