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A High-Dimensional Extension of Wagner's Theorem and the Geometrization of Hypergraphs

Qiming Fang, Sihong Shao

TL;DR

The paper addresses the problem of when a $d$-uniform hypergraph can be embedded into $\mathbb{R}^d$ by geometrizing hyperedges as $(d-1)$-simplices and imposing homotopy constraints via CW complexes. It develops the framework of $\mathbb{R}^d$-hypergraphs, along with bridges, hyper ear decompositions, and $S$-components, to reduce embeddability to a forbidden-minor condition. Under the triangulation requirement and trivial $\pi_i$ for $i\le d-2$, it proves that a triangulated $d$-uniform topological hypergraph embeds in $\mathbb{R}^d$ iff it contains neither $K_{d+3}^d$ nor $K_{3,d+1}^d$ as a minor, thereby generalizing Wagner-type results to higher dimensions. The work introduces a high-dimensional analog of the discharging method within a topological-hypergraph setting and connects embeddability to a Hadwiger-type program, suggesting avenues for high-dimensional chromatic bounds and coloring via embedding theory.

Abstract

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional Euclidean spaces. Under the assumptions of the triangulation and that all $i$-th homotopy groups are trivial for $i \leq d-2$, we provide a necessary and sufficient condition for a $d$-uniform hypergraph to be embeddable in $\mathbb{R}^d$, which can be regarded as a kind of high-dimensional extension of Wagner's Theorem for planar graphs. Specifically, we establish that a triangulated $d$-uniform topological hypergraph embeds into $\mathbb{R}^d$ if and only if it contains neither $K_{d+3}^d$ nor $K_{3,d+1}^d$ as a minor. Here, a triangulated $d$-uniform topological hypergraph constitutes a geometrized form of a $d$-uniform hypergraph, while $K_{d+3}^d$ and $K_{3,d+1}^d$ are the high-dimensional generalizations of the complete graph $K_5$ and the complete bipartite graph $K_{3,3}$ in $\mathbb{R}^d$, respectively.

A High-Dimensional Extension of Wagner's Theorem and the Geometrization of Hypergraphs

TL;DR

The paper addresses the problem of when a -uniform hypergraph can be embedded into by geometrizing hyperedges as -simplices and imposing homotopy constraints via CW complexes. It develops the framework of -hypergraphs, along with bridges, hyper ear decompositions, and -components, to reduce embeddability to a forbidden-minor condition. Under the triangulation requirement and trivial for , it proves that a triangulated -uniform topological hypergraph embeds in iff it contains neither nor as a minor, thereby generalizing Wagner-type results to higher dimensions. The work introduces a high-dimensional analog of the discharging method within a topological-hypergraph setting and connects embeddability to a Hadwiger-type program, suggesting avenues for high-dimensional chromatic bounds and coloring via embedding theory.

Abstract

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional Euclidean spaces. Under the assumptions of the triangulation and that all -th homotopy groups are trivial for , we provide a necessary and sufficient condition for a -uniform hypergraph to be embeddable in , which can be regarded as a kind of high-dimensional extension of Wagner's Theorem for planar graphs. Specifically, we establish that a triangulated -uniform topological hypergraph embeds into if and only if it contains neither nor as a minor. Here, a triangulated -uniform topological hypergraph constitutes a geometrized form of a -uniform hypergraph, while and are the high-dimensional generalizations of the complete graph and the complete bipartite graph in , respectively.

Paper Structure

This paper contains 17 sections, 22 theorems, 14 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Main result and motivation
  3. Comparison with previous research
  4. Two-dimensional closed surfaces
  5. Higher-dimensional Euclidean spaces
  6. The difference between our result and existing ones
  7. Organization of the paper
  8. $\mathbb{R}^d$-hypergraph: Definition and property
  9. Closed $\mathbb{R}^d$-hypergraph and triangulated $\mathbb{R}^d$-hypergraph ($d\geq 3$)
  10. Bridges
  11. Hyper ear decomposition
  12. Recognizing $\mathbb{R}^d$-hypergraph
  13. $S$-component
  14. Connectivity
  15. Anti-$d$-dimension minor
  16. ...and 2 more sections

Key Result

Theorem 1.1

A triangulated $d$-uniform topological hypergraph embeds into $\mathbb{R}^d$ if and only if it contains neither $K_{d+3}^d$ nor $K_{3,d+1}^d$ as a minor.

Figures (10)

  • Figure 1: The key difference between this paper and previous studies lies in the treatment of $P_2$. Although $P_1$ and $P_2$ share the same $1$-skeleton, namely the complete tripartite graph $K_{2,2,2}$, $P_1$ can be embedded in $\mathbb{R}^3$, while $P_2$ cannot carmesin2023embedding. Thus previous studies regard $P_2$ as a forbidden minor in $\mathbb{R}^3$. Below we are able to show that the triangulation of $P_2$ yields $K_6^3$: First, each of the three squares $x_1x_3x_2x_4$, $x_1x_5x_2x_6$, and $x_3x_5x_4x_6$ must be subdivided into $2$-simplices. Since we focus on simple hypergraphs --- where multiple edges are not exist --- the triangulation must be carried out as follows: (i) Add the edge $x_1x_2$ within $x_1x_3x_2x_4$, subdividing it into two $2$-simplices: $x_1x_2x_3$ and $x_1x_2x_4$; (ii) Add the edge $x_5x_6$ within $x_1x_5x_2x_6$, subdividing it into two $2$-simplices: $x_1x_5x_6$ and $x_2x_5x_6$; (iii) Add the edge $x_3x_4$ within $x_3x_5x_4x_6$, subdividing it into two $2$-simplices: $x_3x_4x_5$ and $x_3x_4x_6$. Next, we further add the $2$-simplices $x_1x_2x_5$, $x_1x_2x_6$, $x_3x_5x_6$, $x_4x_5x_6$, $x_1x_3x_4$, and $x_2x_3x_4$ to $P_2$. Let $P_2^\Delta$ denote the resulting triangulated complex. It is straightforward to verify that every polyhedral $3$-cell in $P_2^\Delta$ is a tetrahedron. Moreover, for any three vertices $\{x_i, x_j, x_k\}\subseteq \{x_1, x_2, x_3, x_4, x_5, x_6\}$, there exists a $2$-simplex $x_ix_jx_k$ in $P_2^\Delta$. Therefore, $P_2^\Delta$ corresponds precisely to $K_6^3$ as described in Theorem \ref{['anti-minor']}.
  • Figure 2: Ear decomposition.
  • Figure 3: An $\mathbb{R}^3$-hypergraph without pendant simplex.
  • Figure 4: $B_1$ and $B_2$ are skew of $S^2$.
  • Figure 5: An example of $S$-decomposition and marked $S$-decomposition of $\mathbb{R}^3$-hypergraph.
  • ...and 5 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 1: faces of a $k$-simplex
  • Definition 2: $d$-uniform-topological hypergraph
  • Definition 3: induced sub-$d$-uniform-topological hypergraph
  • Definition 4: homotopy group constraint
  • Definition 5: $\mathbb{R}^d$-hypergraph and non-$\mathbb{R}^d$-hypergraph
  • Lemma 2.1: Construction of $\mathbb{R}^d$-hypergraph
  • proof
  • Lemma 2.2
  • proof
  • ...and 57 more