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Edge isoperimetry of lattices

Cameron Strachan, Konrad Swanepoel

TL;DR

This work studies the edge isoperimetric problem (EIP) on Cayley graphs of $\mathbb{Z}^d$, focusing on the existence of nested sequences of optimal $n$-vertex subsets. It delivers a negative result for all $d\ge2$ by constructing a specific Cayley graph on $\mathbb{Z}^d$ that admits no nested ordering, using a Loomis–Whitney bound to force containment in a finite box and deriving a contradiction with a known grid-bound. It also presents a positive two-dimensional example on a unit-length triangular lattice (isomorphic to $\mathbb{Z}^2$) that does admit a nested ordering, proven through a detailed hull-analysis of 12-direction polygons and a computational inductive scheme. Together, these results illustrate that nested EIP solutions can fail in higher dimensions while existing in meaningful 2D lattice settings, and they outline computationally aided strategies that may extend to broader generating sets.

Abstract

We present two results related to an edge-isoperimetric question for Cayley graphs on the integer lattice asked by Ben Barber and Joshua Erde [Isoperimetry of Integer Lattices, Discrete Analysis 7 (2018)]. For any (undirected) graph $G$, the edge boundary of a subset of vertices $S$ is the number of edges between $S$ and its complement in $G$. Barber and Erde asked whether for any Cayley graph on $\mathbb{Z}^d$, there is always an ordering of $\mathbb{Z}^d$ such that for each $n$, the first $n$ terms minimize the edge boundary among all subsets of size $n$. First, we present an example of a Cayley graph $G_d$ on $\mathbb{Z}^d$ (for all $d\geq 2$) for which there is no such ordering. Furthermore, we show that for all $n$ and any optimal $n$-vertex subset $S_n$ of $G_d$, there is no infinite sequence $S_n\subset S_{n+1}\subset S_{n+2}\subset\cdots$ of optimal sets $S_i$, where $|S_i|=i$ for $i\geq n$. This is to be contrasted with the positive result in $\mathbb{Z}^1$ shown by Joseph Briggs and Chris Wells [arXiv:2402.14087]. Our second result is a positive example for the unit-length triangular lattice (which is isomorphic to $\mathbb{Z}^2$) where two vertices are connected by an edge if their distance is $1$ or $\sqrt{3}$. We show that this graph has such an ordering. This is the most complicated example known to us of a two-dimensional Cayley graph for which an ordering exists.

Edge isoperimetry of lattices

TL;DR

This work studies the edge isoperimetric problem (EIP) on Cayley graphs of , focusing on the existence of nested sequences of optimal -vertex subsets. It delivers a negative result for all by constructing a specific Cayley graph on that admits no nested ordering, using a Loomis–Whitney bound to force containment in a finite box and deriving a contradiction with a known grid-bound. It also presents a positive two-dimensional example on a unit-length triangular lattice (isomorphic to ) that does admit a nested ordering, proven through a detailed hull-analysis of 12-direction polygons and a computational inductive scheme. Together, these results illustrate that nested EIP solutions can fail in higher dimensions while existing in meaningful 2D lattice settings, and they outline computationally aided strategies that may extend to broader generating sets.

Abstract

We present two results related to an edge-isoperimetric question for Cayley graphs on the integer lattice asked by Ben Barber and Joshua Erde [Isoperimetry of Integer Lattices, Discrete Analysis 7 (2018)]. For any (undirected) graph , the edge boundary of a subset of vertices is the number of edges between and its complement in . Barber and Erde asked whether for any Cayley graph on , there is always an ordering of such that for each , the first terms minimize the edge boundary among all subsets of size . First, we present an example of a Cayley graph on (for all ) for which there is no such ordering. Furthermore, we show that for all and any optimal -vertex subset of , there is no infinite sequence of optimal sets , where for . This is to be contrasted with the positive result in shown by Joseph Briggs and Chris Wells [arXiv:2402.14087]. Our second result is a positive example for the unit-length triangular lattice (which is isomorphic to ) where two vertices are connected by an edge if their distance is or . We show that this graph has such an ordering. This is the most complicated example known to us of a two-dimensional Cayley graph for which an ordering exists.

Paper Structure

This paper contains 12 sections, 6 theorems, 38 equations, 13 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Non-existence of an ordering
  3. Proof outline of Theorem \ref{['the:e']}
  4. The upper bound for Theorem \ref{['the:e']}
  5. The upper bound for polygons
  6. Proof of Theorem \ref{['the:P']}
  7. Base Cases
  8. Inductive step
  9. Remaining cases in the inductive step
  10. Proof of Upper Bound of Theorem \ref{['the:e']}
  11. Existence of an ordering
  12. Conclusion

Key Result

Theorem 1

The EIP for $\mathbb{Z}^d_U$, where $U$ is the generating set $\{\pm e_i : i=1,\dots,d\}\cup\{\pm 2e_1\}$ of $\mathbb{Z}^d$, does not have nested solutions starting at any size. In other words, for all $n$ and each $n$-element subset $S_n$ of $\mathbb{Z}^d$ for which $\partial(S_n)$ is the minimum a

Figures (13)

  • Figure 1: Generating set of Theorem \ref{['the:e']}
  • Figure 2: The extremal subgraph of $\Lambda_U$ with $24k^2-24k+7$ vertices ($k=2$)
  • Figure 3: The hull of the black points is the set of black and red points
  • Figure 4: What sides of $P$ correspond to the parameters $u_i$ and $t_i$$(i=1,\dots, 6)$
  • Figure 5: Circumscribed parallelogram (blue) and hexagon (red) of $P$ (gray)
  • ...and 8 more figures

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Conjecture 3
  • proof : Proof of Theorem \ref{['the:']}
  • Definition 3
  • Definition 4
  • Theorem 4
  • Lemma 5
  • ...and 28 more