Hamilton Cycles and Paths in the Noncrossing Partition Refinement Graph

Let NC_R(n) be the cover graph of the refinement order on the noncrossing partitions of an n-element cyclically ordered set. Thus an edge merges two blocks, or reverses such a merge by splitting one block. We determine exactly when this graph has a Hamilton cycle. For odd n ≥ 3, block-count parity is a bipartition and a signed Dyck-tree recurrence gives color-class difference (-1)^(m+1) Cat_m at n = 2m + 1, so a Hamilton cycle is impossible. For every even n ≥ 4, we give one uniform construction. A noncrossing partition is encoded bijectively by a noncrossing partial matching Q and a Boolean mask on the unmatched nonroot points. Each fixed Q is therefore an odd-dimensional hypercube and carries a cyclic reflected Gray code. Deleting the lexicographically first chord makes the matchings into a rooted tree. For every tree edge we exhibit an explicit refinement diamond; an injective port assignment makes all selected directed cube-edge occurrences distinct. Recursive square switching then joins all Boolean cycles into one Hamilton cycle. If a one-vertex graph is regarded as Hamiltonian via its constant closed spanning walk, the complete classification for all natural n is: NC_R(n) is Hamiltonian if and only if n ∈ {0, 1}, or n is even and n ≥ 4.