Note on disjoint faces in simple topological graphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912131278962688 |
|---|---|
| author | Zeng, Ji |
| author_facet | Zeng, Ji |
| contents | We prove that every $n$-vertex complete simple topological graph generates at least $Ω(n)$ pairwise disjoint $4$-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every $n$-vertex complete simple topological graph drawn in the unit square generates a $4$-face with area at most $O(1/n)$. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for $k$-faces with arbitrary $k\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04742 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Note on disjoint faces in simple topological graphs Zeng, Ji Combinatorics We prove that every $n$-vertex complete simple topological graph generates at least $Ω(n)$ pairwise disjoint $4$-faces. This improves upon a recent result by Hubard and Suk. As an immediate corollary, every $n$-vertex complete simple topological graph drawn in the unit square generates a $4$-face with area at most $O(1/n)$. This can be seen as a topological variant of the Heilbronn problem for quadrilaterals. We construct examples showing that our result is asymptotically tight. We also discuss the similar problem for $k$-faces with arbitrary $k\geq 3$. |
| title | Note on disjoint faces in simple topological graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2308.04742 |