On minimum Venn diagrams

TL;DR

The work tackles the minimum-crossings problem for $n$-Venn diagrams by constructing near-minimum diagrams via combinatorial duals on the hypercube. The authors combine isometric partitions of the hypercube with a long-run Gray code to minimize faces in the dual plane graph, achieving an $8$-Venn diagram with $40$ crossings and, for $n=2^k$ with $k\ge 4$, diagrams with at most $(1+\frac{33}{8n})L_n$ crossings, plus a doubling construction to extend to nearby $n$ with $(2+o(1))L_{n+m}$. These contributions yield the smallest-known crossing counts for all $n\ge 8$ and provide a scalable asymptotic framework based on dual hypercube graphs and Ramras-inspired partitions. The results illuminate the structure of minimum crossing arrangements via dual hypercube graphs and open questions about exact existence for all $n$ and potential improvements in the asymptotic constants.

Abstract

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.

Figures (11)

• Figure 1: Simple Venn diagrams for $n=3,4,5$, maximizing the number of crossings. For $n=3,4,5$ the number of crossings $2^n-2$ equals $6,14,30$. The diagrams (a)+(c) are rotationally symmetric. All diagrams are monotone and convex.
• Figure 2: Non-simple Venn diagrams for $n=3,4,5,6,7$ that minimize the number of crossings. For those values of $n$, the number of crossings $L_n=\lceil\frac{2^n-2}{n-1}\rceil$ equals $3,5,8,13,21$. Only in the diagrams (a) and (e) every crossing involves all $n$ curves. The diagrams (a) and (e) are symmetric. The diagram (a) is monotone, while (b)--(e) are not monotone. None of the diagrams is convex, though (a) can be made convex by changing the radii of the circular arcs.
• Figure 3: An 8-Venn diagram with 40 crossings obtained from Theorem \ref{['thm:min']}. The 8 curves are drawn with colors. Each black/white bubble represents one crossing that has to be contracted to a single point, where the open bubbles at the left and right boundary wrap around at the bottom as if drawn on a cylinder.
• Figure 4: Log-linear plot of the ratio $r$ of the number of crossings of the Venn diagrams obtained from Theorem \ref{['thm:min-all']} and the lower bound.
• Figure 5: The dual graph of the 4-Venn diagram from Figure \ref{['fig:dia2']} (b) is a spanning subgraph of the 4-cube.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Theorem 3: MR1152451
• Lemma 4
• proof
• Lemma 5
• proof
• Corollary 6
• Lemma 7
• Lemma 8