发表机构
School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Berge猜想对无桥无爪三次图成立，并给出四个完美匹配覆盖至少299/315边数的结果。
AI 中文摘要
Berge猜想[Proc. London Math. Soc., 1979]断言每个无桥三次图可以被五个完美匹配覆盖。5-圈双覆盖猜想由Preissmann(1981)和Celmins(1984)独立提出，断言每个无桥图允许五个偶子图，使得每条边恰好属于其中两个。Hakobyan和Mkrtchyan[Ars Math. Contemp., 2019]证明了5-圈双覆盖猜想成立当且仅当每个无桥无爪三次图可以被四个完美匹配覆盖。在本文中，我们证明了Berge猜想对无桥无爪三次图成立。此外，我们证明每个无桥无爪三次图$G$允许四个完美匹配覆盖至少$\frac{299}{315}|E(G)|$条边。
英文摘要
The Berge conjecture [Proc. London Math. Soc., 1979] asserts that every bridgeless cubic graph can be covered by five perfect matchings. The 5-cycle double cover conjecture, proposed independently by Preissmann (1981) and Celmins (1984), asserts that every bridgeless graph admits five even subgraphs such that every edge belongs to exactly two of them. Hakobyan and Mkrtchyan [Ars Math. Contemp., 2019] proved that the 5-cycle double cover conjecture holds if and only if every bridgeless claw-free cubic graph can be covered by four perfect matchings. In this paper, we prove that the Berge conjecture holds for bridgeless claw-free cubic graphs. Moreover, we show that every bridgeless claw-free cubic graph $G$ admits four perfect matchings that cover at least $\frac{299}{315}|E(G)|$ edges.
