Tearing a strip off the plane
Tearing a strip off the plane
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从飞机上撕下一条条
作者：
B. Bauslaugh
在本文中，我们研究了单位距离图的某些子图的同态性质。我们定义Gr是由平面的子集(-∞，∞)×[0，r]导出的单位距离图的子图。本文的主要内容是研究图Gr，其中r是使Gr包含给定长度的奇圈的最小宽度。对于每个奇数n，我们确定了使得$G_{r_{n}}$包含n-圈Cn的最小宽度Rn，并刻划了Cn在$G_{r_{n}}$中的嵌入。证明了当n≡3(Mod 4)时，$G{r{n}}$与Cn同态等价；而当n≡1(Mod 4)时，$G{r{n}}$是核.我们首先证明了Gr对每个r≥0是同态紧的，如文[1]中所定义的。最后，我们给出了一些与图Gr有关的其他有趣的结果和公开问题。©1998 John Wiley&Sons，Inc.J.图形理论29：17-33,1998
In this article we examine some homomorphic properties of certain subgraphs of the unit-distance graph. We define Gr to be the subgraph of the unit-distance graph induced by the subset (-∞, ∞) × [0, r] of the plane. The bulk of the article is devoted to examining the graphs Gr, when r is the minimum width such that Gr contains an odd cycle of given length. We determine for each odd n the minimum width rn such that $G_{r_{n}}$ contains an n-cycle Cn, and characterize the embeddings of Cn in $G_{r_{n}}$. We then show that $G_{r_{n}}$ is homomorphically equivalent to Cn when n ≡ 3 (mod 4), but $G_{r_{n}}$ is a core when n ≡ 1 (mod 4). We begin by showing that Gr is homomorphically compact for each r ≥ 0, as defined in [1]. We conclude with some other interesting results and open problems related to the graphs Gr. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 17–33, 1998