关于无限无割边立方图中的若干完美匹配猜想
On some perfect matching conjectures in infinite, cubic, bridgeless graphs
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中文总结 AI 辅助
本文研究Berge-Fulkerson等三个关于完美匹配的著名猜想在无限无割边立方图中的有效性，类比有限情形确立其蕴含关系，并证明各猜想的有限与无限版本等价。
中文摘要 AI 辅助
Berge-Fulkerson猜想指出，每个无割边立方图都存在6个完美匹配，使得每条边恰好属于其中2个。该猜想自1971年提出以来一直悬而未决，多年来已提出多个弱化版本，其中最著名的两个是Fan-Raspaud猜想和Máčajová-Škoviera猜想。众所周知，Berge-Fulkerson猜想蕴含Fan-Raspaud猜想，而后者又蕴含Máčajová-Škoviera猜想。这些问题在有限图领域已被研究多年，且已建立起它们与其他结果之间的诸多等价关系，但在无限图领域，人们对它们几乎一无所知。本文研究这些猜想在无限情形下是否仍然成立，类比有限情形确立它们的蕴含关系，并证明它们的有限版本与各自的无限版本等价。
英文摘要
The Berge-Fulkerson Conjecture states that every bridgeless cubic graph has six perfect matchings such that each edge belongs to exactly two of them. This conjecture has remained open since 1971, and several of its weakenings have been proposed over the years. Two of the most prominent are the Fan-Raspaud Conjecture and the Máčajová-Škoviera Conjecture. It is well known that the Berge-Fulkerson Conjecture implies the Fan-Raspaud Conjecture, which in turn implies the Máčajová-Škoviera Conjecture. These problems have been studied for years in the context of finite graphs, and many equivalences between them and other results have been established. However, little to nothing is known about them in the context of infinite graphs. In this paper, we investigate whether these conjectures remain valid in the infinite setting, establish their implications in analogy to the finite case, and prove that their finite versions are equivalent to their respective infinite versions.
