AI 中文总结
研究无零三位流、带标签三角形与无桥多重图的圈双覆盖关系，利用西摩和塔特流定理，将顶点流值与三角形边差关联，通过多种方式证明二元系统可解，进而证明有限无桥多重图有圈双覆盖，解决圈双覆盖猜想。
AI 中文摘要
我们研究了无零三位流、带标签三角形和无桥多重图的圈双覆盖之间的关系。利用西摩和塔特的流定理，我们将每个顶点处的三个流值实现为一个三角形的边差，其边携带\(\F_2^3\)的二元子集。我们将这些局部标签在图边之间的一致性表示为一个二元系统，并通过不一致证书、局部测试奇偶性恒等式和全局双重计数来证明其可解性。由此产生的兼容标签追踪圈组件，其中每条边恰好出现两次，证明了每个有限无桥多重图都有一个圈双覆盖，从而证明了圈双覆盖猜想。
英文摘要
We study the relationship between nowhere-zero three-bit flows, labeled triangles, and cycle double covers of bridgeless multigraphs. Using the flow theorems of Seymour and Tutte, we realize the three flow values at each vertex as the side differences of a triangle whose sides carry two-element subsets of $\F_2^3$. We express agreement of these local labels across graph edges as a binary system and prove its solvability by an inconsistency certificate, a local tester-parity identity, and a global double count. The resulting compatible labels trace cycle components in which every edge occurs exactly twice, proving that every finite bridgeless multigraph has a cycle double cover, proving the cycle double cover conjecture.
