AI 中文总结
本文证明至多有两个2度顶点的奇阶树是边优美图，结合零和块划分与Langford序列完成证明，通过符号验证与大规模实例复核，扩大了反魔法树的范围。
AI 中文摘要
若图G有q条边和p个顶点，当存在从E(G)到{1,…,q}的双射f，使得诱导顶点和f^+(v)（即与顶点v关联的边e的f(e)之和）模p互异时，称G是边优美图。Lee于1989年提出猜想：所有奇阶树都是边优美的；目前已知最宽泛的一般性结果（等价形式由Kaplan、Lev和Roditty给出）覆盖至多有1个2度顶点的奇阶树。本文证明：所有至多有2个2度顶点的奇阶树都是边优美的。该证明结合了Zₙ的零和块划分与完美及钩状Langford序列；对所有奇数n≤5001，剩余的情形分析已通过符号验证，且对更大的奇数n也成立；该构造在所有2245070棵阶数不超过25且恰有2个2度顶点的奇阶树上已执行并独立复核。由于边优美图必为反魔法图，该定理也扩大了已知为反魔法图的树的范围。
英文摘要
A graph G with q edges and p vertices is edge-graceful if some bijection f from E(G) onto {1,...,q} makes the induced vertex sums f^+(v), the sum of f(e) over the edges e incident to v, distinct modulo p. Lee conjectured in 1989 that every tree of odd order is edge-graceful; the broadest general result we have located, due in equivalent form to Kaplan, Lev and Roditty, covers trees of odd order with at most one vertex of degree two. We prove that every tree of odd order with at most two vertices of degree two is edge-graceful. The proof combines zero-sum block partitions of Z_n with perfect and hooked Langford sequences; the residual case analysis is verified symbolically for all odd n <= 5001 and holds uniformly beyond, and the construction was executed and independently re-checked on all 2,245,070 trees of odd order at most 25 with exactly two vertices of degree two. Since an edge-graceful graph is antimagic, the theorem also enlarges the family of trees known to be antimagic.
Comments7 pages. Ancillary files contain the verification scripts; the full supplementary archive, including per-tree certificates, is at doi:10.5281/zenodo.22085671
