A note on lenses in arrangements of pairwise intersecting circles in the plane
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916152356110336 |
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| author | Pinchasi, Rom |
| author_facet | Pinchasi, Rom |
| contents | Let $\F$ be a family of $n$ pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by $\F$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in $\F$ touch, then the geometric graph $G$ on the set of centers of the circles in $\F$ whose edges correspond to the lenses generated by $\F$ does not contain pairs of avoiding edges. That is, $G$ does not contain pairs of edges that are opposite edges in a convex quadrilateral. Such graphs are known to have at most $2n-2$ edges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05270 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on lenses in arrangements of pairwise intersecting circles in the plane Pinchasi, Rom Combinatorics Let $\F$ be a family of $n$ pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by $\F$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in $\F$ touch, then the geometric graph $G$ on the set of centers of the circles in $\F$ whose edges correspond to the lenses generated by $\F$ does not contain pairs of avoiding edges. That is, $G$ does not contain pairs of edges that are opposite edges in a convex quadrilateral. Such graphs are known to have at most $2n-2$ edges. |
| title | A note on lenses in arrangements of pairwise intersecting circles in the plane |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.05270 |