关于具有指定平均度和交叉数的交叉-临界图无限族的技术细节
Technical details regarding infinite families of crossing-critical graphs with prescribed average degree and crossing number
摘要：
Siran构造了无限族的k-交叉-临界图的每一个k > 2和Kochol构造了这样的家庭的单图的每一个k > 1。Richter和Richassen认为，对于任意给定的k > 0和r > 5，只有1000个简单的k-交叉-临界图具有最小度r。Salazar观察到，同样的论证意味着对于规定的平均度r > 6的简单k-交叉-临界图也有这样的结论。他建立了存在无限家庭的简单k-交叉-临界图与任何规定的合理平均度r在[4，6）无限多k，并要求他们的存在r在（3，4）。Pinontoan和Richter部分地解决了这个问题，他们在（3.5，4）中对r作了肯定的回答。目前的贡献使用两个新的建设交叉临界单图随着一个开发的Pinontoan和Richter统一这些结果，并回答Salazar的问题，由以下声明：存在无限家庭的简单k-交叉-临界图与任何规定的平均度r（3，6），任何k大于一些下界N（r）。此外，k的一个普遍下界N（I）适用于（3，6）中包含的任何闭区间I中的有理数。这本Mathematica笔记本是作为上述结果的补充。它包含了论文中省略的技术细节，可以作为如何严格验证论文主要结构中参数约束的提示。读者可以使用Mathematica或其他软件来验证所列的权利要求，或者可以以更清晰，老式的方式推导它们。由于笔记本不是独立的，建议在阅读之前阅读论文。
英文摘要：
Siran constructed infinite families of k-crossing-critical graphs for every k > 2 and Kochol constructed such families of simple graphs for every k > 1. Richter and Thomassen argued that, for any given k > 0 and r > 5, there are only finitely many simple k-crossing-critical graphs with minimum degree r. Salazar observed that the same argument implies such a conclusion for simple k-crossing-critical graphs of prescribed average degree r > 6. He established the existence of infinite families of simple k-crossing-critical graphs with any prescribed rational average degree r in [4, 6) for infinitely many k and asked about their existence for r in (3, 4). The question was partially settled by Pinontoan and Richter, who answered it positively for r in (3.5, 4). The present contribution uses two new constructions of crossing critical simple graphs along with the one developed by Pinontoan and Richter to unify these results and to answer Salazar's question by the following statement: there exist infinite families of simple k-crossing-critical graphs with any prescribed average degree r in (3, 6), for any k greater than some lower bound N(r). Moreover, a universal lower bound N(I) on k applies for rational numbers in any closed interval I contained in (3, 6). This Mathematica notebook is presented as a supplement of the paper with the aforementioned results. It contains technical details omitted in the paper and can be used as a hint of how to rigorously verify the constraints that are imposed on the parameters in the main construction of the paper. The reader may either use Mathematica or some other software to verify the listed claims, or may derive them in a more clear, oldfashioned way. As the notebook is not self-contained, it is advisable to read the paper before.
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。
