Hamilton Cycles in the Noncrossing Partition Refinement Graph

This deposit accompanies the paper "Hamilton Cycles in the Noncrossing Partition Refinement Graph" by Alex Chengyu Li. We introduce NCR (n), the graph whose vertices are all noncrossing partitions of n and whose edges connect partitions related by a single block split or merge. We prove that no Hamilton cycle exists when n is odd, via a new short proof of the Narayana parity identity, and construct Hamilton cycles computationally for every even n from 4 to 14 (up to 2, 674, 440 vertices). We also establish that the minimum degree equals n − 1, attained by exactly n vertices with a two-chain structure. We conjecture that a Hamilton cycle exists for every even n ≥ 4. Deposit contents: Compiled PDF, Python verification scripts, pre-computed Hamilton cycle data for n = 4. . 14, and an interactive proof explorer (HTML). Code and live explorer: https: //github. com/crabsatellite/gray-code-evolution Changes in v2: Added minimum degree theorem with full proof (splits bound, merge-graph connectivity, uniqueness characterization). Added interactive proof explorer. Expanded cross-chain merge argument. Improved exposition throughout.