AI 中文总结
该研究证明度至少7的正则图的控制数不超过边控制数，填补度7、8的空白，立方图等中间度情况仍未解决。
AI 中文摘要
Baste等人（2020）提出猜想：每个正度正则图的控制数不超过其边控制数，边控制数即最大匹配的最小规模。结合已发表的界，该不等式对所有度至少为9的图成立。一种归约方法证明：当最小最大匹配的每条边的一个端点可被选作控制集时，该不等式成立；而Lovász局部引理表明，对所有度至少为7的图，均存在这样的选择，这一新结果填补了度7和度8的空白，仅度3至度6的情况仍未解决。该归约方法可对每个未解决的度，在顶点数不超过某一有界值时解决，其中立方图（3-正则图）的该值为48。不过，在顶点数为50时，该归约方法遇到了一个具体的立方图无法解决该问题，但该图的不等式依然成立。此外，该不等式无法进一步收紧，因为存在无穷多个立方图，其控制数与边控制数相等。立方图的情况仍未解决，即使是基于局部结构的线性论证也无法解决它；中间度的情况，除已解决的图外，其余仍未解决。
英文摘要
Baste et al. (2020) conjectured that every regular graph of positive degree has domination number at most its edge domination number, the least size of a maximal matching. Combining published bounds settles the inequality for every degree at least nine. A reduction proves the inequality whenever one endpoint of each edge of a minimum maximal matching can be chosen to form a dominating set, and the Lovász Local Lemma shows such a choice exists for every degree at least seven, newly closing degrees seven and eight and leaving degrees three through six open. The reduction settles each open degree up to a bounded number of vertices, forty-eight for cubic graphs. At fifty vertices, however, the reduction meets an explicit cubic graph it cannot settle, though the inequality holds there too. The inequality cannot be tightened, since infinitely many cubic graphs have equal domination and edge domination numbers. The cubic case stays open, and even linear arguments from the local structure cannot close it. The middle degrees stay open beyond the graphs already settled.
Comments15 pages, 1 table
