多重孤立核与树的色对称重构
Multiplicity-Isolated Cores and Chromatic Symmetric Reconstruction of Trees
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中文总结 AI 辅助
该研究针对Stanley树同构猜想，提出允许重复叶分量阶的树重构准则，证明一类无限直径6树的色对称函数可重构，还给出相关划分模型与实现。
中文摘要 AI 辅助
Stanley的树同构猜想询问色对称函数是否能区分非同构的树。我们给出了允许重复叶分量阶的重构准则：对于一棵真树，将每个叶分量收缩到其中心，并将其阶记录为顶点权重。我们证明，当该加权核的每个非叶顶点的权重在核中其他地方都不出现时，色对称函数可重构该树，而核叶之间的重复是不受限制的。该证明仅使用了星基划分及其正上方的系数。由此，该猜想对一类无限的直径为6的树成立，这类树不满足所有叶分量阶均不同的条件。我们还为任意直径为6的树给出了一个规范整数划分模型，以及一个用于后续研究的精确割划分实现，而不受限制的直径为6的情况仍待解决。
英文摘要
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.
