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The aim is to determine the conditions under which the noncrossing partition refinement graph has a Hamilton cycle.

July 14, 2026Open Access

Hamilton Cycles in the Noncrossing Partition Refinement Graph

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The aim is to determine the conditions under which the noncrossing partition refinement graph has a Hamilton cycle.
Analyzed the cover graph of noncrossing partitions NC_R(n) for n-element cyclically ordered sets.
Determined Hamiltonian conditions based on block-count parity for odd and even n.
Used a bijective encoding with a noncrossing partial matching and Boolean masks.
For odd n (n >= 3), a Hamilton cycle is impossible due to block-count parity.
For even n (n >= 4), provided a uniform construction leads to a Hamilton cycle.
Complete classification indicates NC_R(n) is Hamiltonian if and only if n is 0, 1, or even and >= 4.

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Randomized trial shows conditions for Hamilton cycles in noncrossing partition graphs, indicating structural limitations.

Alex Li (2026) studied this question.

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