3-空间中具有同伦自由外部的线性嵌入图
Linearly embedded graphs in 3-space with homotopically free exteriors
摘要：
如果图的任何边被发送为线段，则将图嵌入到$\mathbb{R}^3$中被称为是线性的。如果$π_1(\mathbb{R}^3-f(G))$是一个自由群，则称图$G$到$\mathbb{R}^3$的嵌入$f$是自由的.对于任何完全图，其线性嵌入总是自由的。本文研究了线性嵌入的自由度问题。证明了对于任意至多6个顶点的简单连通图，如果它的最小价至少为3，则它的线性嵌入总是自由的。相反，当顶点数远大于最小价数或连通度时，自由度可能不是这类图的固有性质。事实上，我们证明了对于任何$n \geq 1$，存在无穷多个具有最小价$n$的连通图具有非自由线性嵌入，并且存在无穷多个具有非自由线性嵌入的$n$ -连通图.
英文摘要：
An embedding of a graph into $\mathbb{R}^3$ is said to be linear, if any edge of the graph is sent to be a line segment. And we say that an embedding $f$ of a graph $G$ into $\mathbb{R}^3$ is free, if $π_1(\mathbb{R}^3-f(G))$ is a free group. It was known that for any complete graph its linear embedding is always free. In this paper we investigate the freeness of linear embeddings considering the number of vertices. It is shown that for any simple connected graph with at most 6 vertices, if its minimal valency is at least 3, then its linear embedding is always free. On the contrary when the number of vertices is much larger than the minimal valency or connectivity, the freeness may not be an intrinsic property of such graphs. In fact we show that for any $n \geq 1$ there are infinitely many connected graphs with minimal valency $n$ which have non-free linear embeddings, and furthermore, that there are infinitely many $n$-connected graphs which have non-free linear embeddings.
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