无桥无爪三次图中的完美匹配三元组
A tris of perfect matchings in bridgeless claw-free cubic graphs
- L-Università ta’ Malta（马耳他大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明了Fan--Raspaud猜想对无桥无爪三次图成立，即存在三个完美匹配交集为空，并提供了与5-圈双覆盖猜想相关的信息。
中文摘要 AI 辅助
最近有人宣布证明了圈双覆盖猜想，从而得出每个无桥图都有一个$8$-圈双覆盖。更强的$5$-圈双覆盖猜想仍然开放，它等价于以下陈述：每个无桥无爪三次图的边集可以被至多四个完美匹配覆盖。无桥三次图中的完美匹配已被广泛研究，该领域的两个主要猜想是Berge--Fulkerson猜想和Fan--Raspaud猜想。后者是前者的推论，它断言每个无桥三次图都存在三个完美匹配$M_1, M_2, M_3$，使得$M_1\cap M_2\cap M_3=\emptyset$。在这里我们证明Fan--Raspaud猜想对无桥无爪三次图成立。这也为在$5$-圈双覆盖猜想需要控制四个完美匹配的类别中完美匹配的相互作用提供了进一步的信息。
英文摘要
A proof of the cycle double cover conjecture was recently announced, yielding an $8$-cycle double cover for every bridgeless graph. The stronger $5$-cycle double cover conjecture, which is still open, is equivalent to the statement that the edge set of every bridgeless claw-free cubic graph can be covered by at most four perfect matchings. Perfect matchings in bridgeless cubic graphs have been studied extensively, with two of the main conjectures in this area being the Berge--Fulkerson and the Fan--Raspaud conjectures. The latter, a consequence of the former, states that every bridgeless cubic graph admits three perfect matchings $M_1, M_2, M_3$ such that $M_1\cap M_2\cap M_3=\emptyset$. Here we show that the Fan--Raspaud conjecture is true for bridgeless claw-free cubic graphs. This also gives further information on the interaction of perfect matchings in a class where the $5$-cycle double cover conjecture requires control of four of them.
