Hamilton Cycles and Paths in the Noncrossing Partition Refinement Graph

Let NCR(n) be the cover graph of the refinement order on the noncrossing partitions of an n-element cyclically ordered set. We determine exactly when this graph has a Hamilton cycle and when it has a Hamilton path. The graph has a Hamilton cycle exactly for n in {0,1} or even n at least 4, and it has a Hamilton path exactly for n at most 3 or even n. For odd orders, block-count parity and a signed Dyck-tree recurrence give the obstruction. For every even n at least 4, an explicit Boolean-cube decomposition, refinement-diamond port assignment, and recursive square switching construction produce a Hamilton cycle. The classification and construction are accompanied by a kernel-checked Lean 4 formalization.