| Autor: |
Remie Janssen, Leonie van Steijn |
| Jazyk: |
angličtina |
| Rok vydání: |
2022 |
| Předmět: |
|
| Zdroj: |
Theory and Applications of Graphs, Vol 9, Iss 1 (2022) |
| Druh dokumentu: |
article |
| ISSN: |
2470-9859 |
| DOI: |
10.20429/tag.2022.090102 |
| Popis: |
The unit distance graph $G^1_{R^d}$ is the infinite graph whose nodes are points in $R^d$, with an edge between two points if the Euclidean distance between these points is $1$. The 2-dimensional version $G^1_{R^2}$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G^1_{R^d}$ to closed convex subsets $X$ of $R^d$. We show that the graph $G^1_{R^d}[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to $0$, or when $r(X)\geq 1$ and the affine dimension of $X$ is at least $2$. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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