A uniquely pancyclic graph is a graph that has exactly one cycle of each length between 3 and the graph's vertex count. Uniquely pancyclic graphs are therefore a
special case of pancyclic graphs.
By their definition, uniquely pancyclic graphs are also uniquely HamiltonianUniquely Hamiltonian Graph.
Determining which simple graphs are uniquely pancyclic is an open problem asked by Roger Entringer in 1973 and posed by Bondy and Murty (1976, Problem 10, p. 247). It is conjectured that the 7 graphs illustrated above (Shi 1986) are the only uniquely pancyclic graphs, but proof or discovery of additional examples have thus far remained elusive.
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