Multiplicity-Isolated Cores and Chromatic Symmetric Reconstruction of Trees

Zijian Zeng

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arXiv · 2608.18179

Multiplicity-Isolated Cores and Chromatic Symmetric Reconstruction of Trees

Zijian Zeng

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Abstract

Stanley's tree-isomorphism conjecture asks whether the chromatic symmetric function distinguishes nonisomorphic trees. We give a reconstruction criterion that permits repeated leaf-component orders. For a proper tree, collapse each leaf component to its center and record its order as a vertex weight. We prove that the chromatic symmetric function reconstructs the tree whenever every nonleaf vertex of this weighted core has a weight that occurs nowhere else in the core. Repetitions among core leaves are unrestricted. The proof uses only the leading star-basis partition and the coefficients immediately above it. As a consequence, the conjecture holds for an infinite class of diameter-six trees not covered by the condition that all leaf-component orders are distinct. We also give a canonical integer-partition model for arbitrary diameter-six trees and an exact cut-partition implementation intended for further work. The unrestricted diameter-six case remains open.

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Zijian Zeng. 2026-08-18. Multiplicity-Isolated Cores and Chromatic Symmetric Reconstruction of Trees. https://arxiv.org/abs/2608.18179

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