关于三次图多面体嵌入的亏格
On the genera of polyhedral embeddings of cubic graph
AI总结：
本文研究三次图的多面体嵌入，提出高效计算算法，证明偶三次图可在任意高亏格嵌入，但多数三次图无任何亏格的多面体嵌入。
摘要：
在本文中，我们给出了关于图的多面体嵌入存在性的理论和计算结果，重点研究三次图。我们还描述了一种高效算法，用于计算给定三次图的所有多面体嵌入，并构造了具有某些特殊多面体嵌入性质的三次图。一些关键结果是：即使在环面上具有多面体嵌入的偶三次图，也可以在任意高的亏格中具有多面体嵌入，事实上，其亏格可以接近该顶点数目的理论最大值；并且，三次图可以在多少个亏格中具有多面体嵌入没有上限。虽然这些结果表明多面体嵌入具有很大的多样性，但对最多28个顶点的计算表明，绝大多数三次图在任何亏格中都没有多面体嵌入，并且这些图的比例随着顶点数的增加而增加。
英文摘要：
In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.
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。
