A Four-Color-Theorem Proof of the Duplicate-Leaf Parse-Word Conjecture

Cooper, Rowland and Zeilberger conjectured that two ordered full binary trees sharing a pair of consecutive uncle-nephew leaves have a common parse word with equal letters at that prescribed pair. We prove this existence assertion using the four-color theorem. Gluing the two corresponding polygon triangulations gives a planar graph whose specified boundary vertex has degree at most four. Contracting a length-two path, four-coloring the minor, and restoring the vertex forces the desired equal labels. A padding argument, together with the known equivalence for unrestricted parsing, shows that the duplicate-leaf assertion is equivalent to the four-color theorem. This is a proof using that established theorem, not a new proof of it or a four-color-free reduction to the particular smaller tree pair. Exact finite diagnostics and reproducible source are included. This unrefereed preprint makes no absolute novelty claim.