极大无链图的新界
New bounds on maximal linkless graphs
AI总结：
本文构造了新的极大无链可嵌入图族，显著改进了最低边顶比，并给出了团和成为极大无链可嵌入图的充要条件以证明构造的有效性。
摘要：
我们构造了一族在 $n$ 个顶点和 $3n-5$ 条边上的极大无链可嵌入图，适用于所有 $n\ge 10$；以及另一族在 $n$ 个顶点和 $m< \frac{25n}{12}-\frac{1}{4}$ 条边上的图，适用于所有 $n\ge 13$。后者显著改进了任何已知无限族的最低边顶比。我们构造的图族表明，极大无链可嵌入图类不同于 L. Jorgensen 所研究的极大无 $K_6$ 子式的图类。我们给出了两个极大无链可嵌入图在 $K_2$、$K_3$ 或 $K_4$ 上的团和为极大无链可嵌入图的充要条件，并利用这些结果证明我们的构造产生了极大无链可嵌入图。
英文摘要：
We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly improves the lowest edge-to-vertex ratio for any previously known infinite family. We construct a family of graphs showing that the class of maximal linklessly embeddable graphs differs from the class of graphs that are maximal without a $K_6$ minor studied by L. Jorgensen. We give necessary and sufficient conditions for when the clique sum of two maximal linklessly embeddable graphs over $K_2$, $K_3$, or $K_4$ is a maximal linklessly embeddable graph, and use these results to prove our constructions yield maximal linklessly embeddable graphs.
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。
