Did you know that two different graphs with 81 vertices and 20 edges/vertex are famous enough to have Wikipedia articles?
The strongly regular Brouwer–Haemers graph (https://en.wikipedia.org/wiki/Brouwer%E2%80%93Haemers_graph) connects elements of GF(81) that differ by a fourth power.
The Sudoku graph (https://en.wikipedia.org/wiki/Sudoku_graph) connects cells of a Sudoku grid that should be unequal. Sudoku puzzles are instances of precoloring extension on this graph.
Unfortunately the natural graphs on the 81 cards of Set have degree ≠ 20...