树的素数标号
Prime Labelings of Trees
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中文总结 AI 辅助
本文证明了Entringer-Tout猜想：每棵树都存在素数标号，通过结构分解、算术分配和匹配论证，结合计算机辅助证书与解析估计，覆盖所有阶数的树。
中文摘要 AI 辅助
图的素数标号是指将图的顶点与从1到顶点数的整数双射对应，使得相邻顶点获得互素的标号。我证明了Entringer-Tout猜想，即每棵树都允许一个素数标号，将Haxell、Pikhurko和Taraz的结果从足够大的树扩展到所有阶数。我的证明结合了树的结构分解、标号的算术组织以及匹配论证。通过分离树的一小部分，根据标号的可整除性分配标号，并应用Hall定理完成标号，我将标号问题归结为一个结构化的二分匹配问题。我使用精确的计算机辅助证书处理有限和中间范围，其有效性在数学上得到证明，而其余范围由解析估计覆盖。这些要素共同为每一阶的树提供了证明。
英文摘要
A prime labeling of a graph labels its vertices bijectively with the integers from one to the number of vertices, so that adjacent vertices receive coprime labels. I prove the Entringer-Tout conjecture that every tree admits a prime labeling, extending the result of Haxell, Pikhurko, and Taraz from sufficiently large trees to all orders. My proof combines structural decompositions of trees, arithmetic organization of the labels, and matching arguments. I reduce the labeling problem to a structured bipartite matching problem by separating a small part of the tree, distributing labels according to their divisibility properties, and applying Hall's theorem to complete the labeling. I handle the finite and intermediate ranges using exact computer-assisted certificates whose validity is justified mathematically, while the remaining range is covered by analytic estimates. Together, these ingredients yield a proof for trees of every order.
发表机构
- HSE University（高等经济大学）
机构由 AI 辅助整理，请以论文原文为准。
