论文标题
无用的立方图中的脱节奇数可以通过单个完美匹配来平息
Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching
论文作者
论文摘要
令$ g $为无用的立方图。 Berge - Fulkerson猜想（1970年代）指出，$ G $承认了六个完美匹配的列表，因此$ g $的每个边缘都完全属于这些完美的匹配中的两个。如果在肯定中得到回答，那么最近的两个猜想也将是正确的：粉丝 - 拉斯波·猜想（1994年），该猜想指出，$ g $允许三个完美的匹配，以至于$ g $的每个边缘最多都属于其中两个； Mazzuoccolo（2013）的猜想指出，$ g $承认两个完美的匹配，其删除产生了$ g $的两部分子图。可以证明，如果$ g $的任意完美匹配，并非总是有可能将其扩展到满足粉丝的陈述的三个或六个完美匹配的列表 - 拉斯帕德和贝尔格 - 福尔克森 - 富克森的猜想。在本文中，我们表明，鉴于任何$ 1^+$ - 因子$ f $（$ g $的子图，其顶点至少具有至少1个）和任意的边缘$ e $ $ g $，始终存在完美的$ m $ g $ co $ g $ co $ e $，例如$ g g \ setMinus（f \ cup cup m M）$ bipArtite。我们的结果意味着Mazzuoccolo的猜想，但不仅是。这也意味着，鉴于$ g $中的任何集合的不交连电电路，都存在$ g $的完美匹配，其中包含本集合中每个电路的至少一个边缘。
Let $G$ be a bridgeless cubic graph. The Berge--Fulkerson Conjecture (1970s) states that $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan--Raspaud Conjecture (1994), which states that $G$ admits three perfect matchings such that every edge of $G$ belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that $G$ admits two perfect matchings whose deletion yields a bipartite subgraph of $G$. It can be shown that given an arbitrary perfect matching of $G$, it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan--Raspaud and the Berge--Fulkerson conjectures, respectively. In this paper, we show that given any $1^+$-factor $F$ (a spanning subgraph of $G$ such that its vertices have degree at least 1) and an arbitrary edge $e$ of $G$, there always exists a perfect matching $M$ of $G$ containing $e$ such that $G\setminus (F\cup M)$ is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in $G$, there exists a perfect matching of $G$ containing at least one edge of each circuit in this collection.
