发表机构
Yonsei University（延世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了具有正规差子群的Cayley有向图中弧不相交哈密顿路径的存在性，给出了精确判据，并证明所有此类图（包括有限阿贝尔群的双生成元Cayley图）均存在两条这样的路径，方法基于哈密顿割谱的距离公式和几何限制定理。
AI 中文摘要
设$G=\langle a,b\rangle$为一个具有不同非单位生成元的有限群，并假设$H=\langle ba^{-1}\rangle$是正规的。我们精确确定了在$H$的非终端陪集上哪些互补生成元选择能够扩展为两条弧不相交的哈密顿路径，且两条路径的终端顶点都在$H$中。相关的返回余数必须避开一个至多包含六个余数的显式集合。当$[G:H]\ge3$时，任意两个非终端陪集可以保持自由，而所有其他选择都被规定；两个是最优的。每个这样的Cayley有向图都有两条弧不相交的哈密顿路径；这包括有限阿贝尔群的所有双生成元Cayley有向图。该判据源自$\mathrm{Cay}(\mathbb{Z}_q;\{D,D+1\})$的哈密顿割谱的完整距离公式。其唯一的非退化例外来自奇数中心系数$(q-1)/2$。几何成分是一个限制定理：两个满足长度不等式的扇形填充整数序列，若其范围重叠且距离至少为2，则其中一个序列包含在另一个序列的单个间隙中。扇形填充限制了交替性，而长度不等式排除了交错。原始射线循环轮廓的自包含证明遵循Curran和Witte的交点几何。
英文摘要
Let $G=\langle a,b\rangle$ be a finite group with distinct nonidentity generators, and suppose that $H=\langle ba^{-1}\rangle$ is normal. We determine exactly which complementary generator choices on the nonterminal cosets of $H$ extend to two arc-disjoint Hamiltonian paths with both terminal vertices in $H$. The associated return residue must avoid an explicit set of at most six residues. When $[G:H]\ge3$, any two nonterminal cosets may be left free while all other choices are prescribed; two is best possible. Every such Cayley digraph has two arc-disjoint Hamiltonian paths; this includes all two-generator Cayley digraphs of finite abelian groups. The criterion follows from a complete distance formula for the Hamiltonian cut spectra of $\mathrm{Cay}(\mathbb{Z}_q;\{D,D+1\})$. Their unique nondegenerate exception comes from the odd central coefficient $(q-1)/2$. The geometric ingredient is a confinement theorem: two sector-filled integer sequences satisfying a length inequality, with overlapping ranges and distance at least two, have one sequence contained in a single gap of the other. Sector filling bounds alternation, and the length inequality excludes interlacing. A self-contained proof of the primitive-ray cycle profile follows the intersection geometry of Curran and Witte.
Comments16 pages, 4 figures. Ancillary files contain the verification programs and explicit path witnesses. Extends the finite abelian case (arXiv:2605.27241) to normal difference subgroups
