What does this research mean for the field?

The cover graph of the refinement order on noncrossing partitions of an n-element cyclically ordered set has a Hamilton cycle exactly for n in {0,1} or even n ≥ 4, and a Hamilton path exactly for n ≤ 3 or even n. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Determine the exact conditions under which the cover graph of the noncrossing partition refinement order, NCR(n), contains a Hamilton cycle or a Hamilton path.

August 27, 2026Open Access

Hamilton Cycles and Paths in the Noncrossing Partition Refinement Graph

View Full Paper

Ask AI

Bookmark

Determine the exact conditions under which the cover graph of the noncrossing partition refinement order, NCR(n), contains a Hamilton cycle or a Hamilton path.
Derived parity obstructions for odd-order graphs using block-count parity and a signed Dyck-tree recurrence relation.
Constructed Hamilton cycles for even orders n ≥ 4 via Boolean-cube decomposition, refinement-diamond port assignment, and recursive square switching.
Formalized and verified all graph classifications and cycle constructions using the Lean 4 interactive theorem prover.
NCR(n) contains a Hamilton cycle if and only if n ∈ {0, 1} or n is an even integer with n ≥ 4.
NCR(n) contains a Hamilton path if and only if n ≤ 3 or n is an even integer.
Odd orders strictly obstruct Hamiltonian cycles and paths due to signed Dyck-tree block-count imbalances.

Discussion

No takes yet. Share an insight, caveat, or question.

Implication

Graph-theoretic analysis characterizes Hamilton cycles and paths in noncrossing partition refinement graphs, demonstrating exact parity conditions and verified Lean 4 constructions.

Cite This Study

Alex Chengyu Li (2026) studied this question.

synapsesocial.com/papers/6a8fea3310c91c1e92622584 https://doi.org/10.5281/zenodo.22098534

View Full Paper

Ask AI

Bookmark

Implication

Graph-theoretic analysis characterizes Hamilton cycles and paths in noncrossing partition refinement graphs, demonstrating exact parity conditions and verified Lean 4 constructions.

Hamilton Cycles and Paths in the Noncrossing Partition Refinement Graph

Key Points

Discussion

Implication

Cite This Study

Hamilton Cycles and Paths in the Noncrossing Partition Refinement Graph

Key Points

Discussion

Implication

Cite This Study

Keep reading

Keep reading