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March 21, 2026Open Access

Hamilton Cycles in the Noncrossing Partition Refinement Graph

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The aim is to explore Hamilton cycles in the noncrossing partition refinement graph NC_R(n) for both odd and even n.
Defined the graph NC_R(n) with noncrossing partitions as vertices.
Proved absence of Hamilton cycles for odd n using the Narayana parity identity.
Constructed Hamilton cycles computationally for even n ranging from 4 to 14.
Established a minimum degree of n − 1 achievable by n vertices in a two-chain structure.
No Hamilton cycle was found for odd n.
Hamilton cycles were computationally generated for every even n from 4 to 14.
Confirmed that n vertices achieve a minimum degree of n − 1.

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The paper demonstrates Hamilton cycles for even n in noncrossing partition graphs, suggesting broader implications.

Li Yelena Alex (2026) studied this question.

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