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arXiv · 2609.31811

The stacking number of a tree

John Fairfax-Ball
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Abstract

The stacking number of a graph is the least integer t >= 2 such that every configuration of t pebbles can be transformed by pebbling moves into a configuration supported on one vertex. We prove that, for every finite tree T with at least two vertices, this number equals the rooted distance-and-degree estimator conjectured by Csernák and Soukup. The proof uses an exact recursive characterization of stackability at a prescribed vertex, an explicit zero-score obstruction, and a weighted cancellation argument for arbitrary nonstackable configurations. The complete theorem is formalized in Lean 4; the formal result has also passed Palomar mechanical verification and is publicly registered as PALOMAR-2026-09-25-000010.

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John Fairfax-Ball. 2026-09-25. The stacking number of a tree. https://arxiv.org/abs/2609.31811

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