Birkhoff Diamond饰演Double Agent
The Birkhoff Diamond as Double Agent
摘要：
尽管存在四色定理的证明，但似乎仍然有更多的东西要了解为什么任何可平面图是4-着色的。为此，我们再看看伯克霍夫钻石，发现一些新的和有趣的：经过广泛的搜索（罕见的）肯普锁定三角形，我们发现一个伯克霍夫钻石子图在每个。我们提供了一个启发式的论点，为什么这一结果不仅是合理的，而且是预期的，并认为伯克霍夫钻石的存在是必要的肯普锁定。如果这个猜想是正确的，这意味着伯克霍夫钻石在4-可着色性问题上扮演着双重角色，同时对给定的可平面图是否可能是最小反例的对边起作用。
英文摘要：
Despite the existence of a proof of the 4-color theorem, it would seem that there is still more to learn about why any planar graph is 4-colorable. To that end, we take another look at the Birkhoff diamond and discover something new and intriguing: after an extensive search for (rare) Kempe-locked triangulations, we find a Birkhoff diamond subgraph in each one. We offer a heuristic argument as to why that result is not only reasonable but also to be expected and posit that the presence of a Birkhoff diamond is necessary to Kempe-locking. If that conjecture is true, it means that the Birkhoff diamond plays a double role in the matter of 4-colorability, simultaneously working for opposite sides of whether a given planar graph could possibly be a minimum counterexample.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
