论文标题
$ r $ - 边缘连接的$ r $ regartular图中的成对脱节完美匹配
Pairwise disjoint perfect matchings in $r$-edge-connected $r$-regular graphs
论文作者
论文摘要
Thomassen [问题1分解常规图时，J。Combin。理论ser。 B，141（2020），343-351]询问是否每$ r $ $连接的$ r $ r $ - 偶数图是否具有$ r-2 $ r-2 $成对的脱节完美匹配。我们表明，如果$ r \ equiv 2 \ text {mod} 4 $，情况并非如此。加上Mattiolo和Steffen的最新结果（高度边缘连接的常规图，没有大型子图，J。Graph Theogry，99（2022），107-116]这解决了Thomassen的所有问题，甚至解决了所有$ r $。事实证明，我们的方法仅限于托马森（Thomassen）问题的情况。然后，我们在高度边缘连接的常规图中证明了对成对分离的完美匹配的一些等效性，其中完美的匹配包含或避免固定的边缘。基于这些结果，我们将5条与5条与5个指标图的成对脱节的完美匹配与众所周知的Cubic图形构成有关，例如Fan-Raspaud猜想，Berge-Fulkerson的猜想和$ 5 $ -Crecycle Double Coventure。
Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is not the case if $r \equiv 2 \text{ mod } 4$. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even $r$. It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the $5$-Cycle Double Cover Conjecture.
