Conway's 99-graph problem

[4][5] The possibility of a graph with these parameters was already suggested in 1969 by Norman L. Biggs,[6] and its existence noted as an open problem by others before Conway.

[3][7][8][9] Conway himself had worked on the problem as early as 1975,[7] but offered the prize in 2014 as part of a set of problems posed in the DIMACS Conference on Challenges of Identifying Integer Sequences.

Other problems in the set include the thrackle conjecture, the minimum spacing of Danzer sets, and the question of who wins after the move 16 in the game sylver coinage.

[1] More generally, there are only five possible combinations of parameters for which a strongly regular graph could exist with each edge in a unique triangle and each non-edge forming the diagonal of a unique quadrilateral.

The 99-graph problem describes the smallest of these combinations of parameters for which the existence of a graph is unknown.