This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The a(4) = 24 graphs:
{12,13,24} {12,13,14,23}
{12,13,34} {12,13,14,24}
{12,14,23} {12,13,14,34}
{12,14,34} {12,13,23,24}
{12,23,34} {12,13,23,34}
{12,24,34} {12,14,23,24}
{13,14,23} {12,14,24,34}
{13,14,24} {12,23,24,34}
{13,23,24} {13,14,23,34}
{13,24,34} {13,14,24,34}
{14,23,24} {13,23,24,34}
{14,23,34} {14,23,24,34}
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[graprms[#]]==n!/2&]],{n,0,5}]
Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 graphs:
{12} {12} {12} {12}
{12,13} {12,13} {12,13}
{12,13,24} {12,13,24}
{12,13,14,23} {12,13,14,23}
{12,13,14,25}
{12,13,24,35}
{12,13,14,23,25}
{12,13,14,23,45}
{12,13,15,24,34}
{12,13,14,15,23,24}
{12,13,14,23,24,35}
{12,13,14,23,25,45}
{12,13,14,15,23,24,35}
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