On minimum Venn diagrams
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2025
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| _version_ | 1866912704454721536 |
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| author | Brenner, Sofia Gregor, Petr Mütze, Torsten Verciani, Francesco |
| author_facet | Brenner, Sofia Gregor, Petr Mütze, Torsten Verciani, Francesco |
| contents | An $n$-Venn diagram is a diagram in the plane consisting of $n$ simple closed curves that intersect only finitely many times such that each of the $2^n$ possible intersections is represented by a single connected region. An $n$-Venn diagram has at most $2^n-2$ crossings, and if this maximum number of crossings is attained, then only two curves intersect in every crossing. To complement this, Bultena and Ruskey considered $n$-Venn diagrams that minimize the number of crossings, which implies that many curves intersect in every crossing. Specifically, they proved that the total number of crossings in any $n$-Venn diagram is at least $L_n:=\lceil\frac{2^n-2}{n-1}\rceil$, and if this lower bound is attained then essentially all $n$ curves intersect in every crossing. Diagrams achieving this bound are called minimum Venn diagrams, and are known only for $n\leq 7$. Bultena and Ruskey conjectured that they exist for all $n\geq 8$. In this work, we establish an asympototic version of their conjecture. For $n=8$ we construct a diagram with 40 crossings, only 3 more than the lower bound $L_8=37$. Furthermore, for every $n$ of the form $n=2^k$ for some integer $k\geq 4$, we construct an $n$-Venn diagram with at most $(1+\frac{33}{8n})L_n=(1+o(1))L_n$ many crossings. Via a doubling trick this also gives $(n+m)$-Venn diagrams for all $0\leq m<n$ with at most $40\cdot 2^m$ crossings for $n=8$ and at most $(1+\frac{33}{8n})\frac{n+m}{n}L_{n+m}=(2+o(1))L_{n+m}$ many crossings for $k\geq 4$. In particular, we obtain $n$-Venn diagrams with the smallest known number of crossings for all $n\geq 8$. Our constructions are based on partitions of the hypercube into isometric paths and cycles, using a result of Ramras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On minimum Venn diagrams Brenner, Sofia Gregor, Petr Mütze, Torsten Verciani, Francesco Combinatorics Discrete Mathematics An $n$-Venn diagram is a diagram in the plane consisting of $n$ simple closed curves that intersect only finitely many times such that each of the $2^n$ possible intersections is represented by a single connected region. An $n$-Venn diagram has at most $2^n-2$ crossings, and if this maximum number of crossings is attained, then only two curves intersect in every crossing. To complement this, Bultena and Ruskey considered $n$-Venn diagrams that minimize the number of crossings, which implies that many curves intersect in every crossing. Specifically, they proved that the total number of crossings in any $n$-Venn diagram is at least $L_n:=\lceil\frac{2^n-2}{n-1}\rceil$, and if this lower bound is attained then essentially all $n$ curves intersect in every crossing. Diagrams achieving this bound are called minimum Venn diagrams, and are known only for $n\leq 7$. Bultena and Ruskey conjectured that they exist for all $n\geq 8$. In this work, we establish an asympototic version of their conjecture. For $n=8$ we construct a diagram with 40 crossings, only 3 more than the lower bound $L_8=37$. Furthermore, for every $n$ of the form $n=2^k$ for some integer $k\geq 4$, we construct an $n$-Venn diagram with at most $(1+\frac{33}{8n})L_n=(1+o(1))L_n$ many crossings. Via a doubling trick this also gives $(n+m)$-Venn diagrams for all $0\leq m<n$ with at most $40\cdot 2^m$ crossings for $n=8$ and at most $(1+\frac{33}{8n})\frac{n+m}{n}L_{n+m}=(2+o(1))L_{n+m}$ many crossings for $k\geq 4$. In particular, we obtain $n$-Venn diagrams with the smallest known number of crossings for all $n\geq 8$. Our constructions are based on partitions of the hypercube into isometric paths and cycles, using a result of Ramras. |
| title | On minimum Venn diagrams |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2511.09230 |