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Discrete isoperimetric inequalities on the strong products of paths

Runze Wang

TL;DR

This paper studies the discrete isoperimetric problem on the strong product $P_x \boxtimes P_y$ of two paths, seeking the minimum vertex boundary size $|\partial_{P_x \boxtimes P_y}(S)|$ for a $k$-set. It develops a compression framework using down- and left-compressions and proves that extremal sets can be taken as compressed and ordered by the box order, yielding explicit lower bounds such as those involving $2\sqrt{k}$, attained by the prefix set $S_0$. The main result (Theorem finite) expresses the minimum as $\min_{i\in \mathcal{A}_k}\{\alpha_i\}$ with intervals $I_i$ and boundary constants $\alpha_i$, and a corollary reduces the problem to $P_x \boxtimes P_{\mathbb N_0}$; both rely on reductions to compressed sets and a five-case analysis. The work discusses limitations with tensor products due to disconnection and outlines extensions to higher dimensions as future directions.

Abstract

For a graph $G=(V,\ E)$ and a nonempty set $S\subseteq V$, the \emph{vertex boundary} of $S$, denoted by $\partial_G(S)$, is defined to be the set of vertices that are not in $S$ but have at least one neighbor in $S$. In this paper, for $G$ being a strong product of two paths, we determine the cases in which $|\partial_G(S)|$ is minimized.

Discrete isoperimetric inequalities on the strong products of paths

TL;DR

This paper studies the discrete isoperimetric problem on the strong product of two paths, seeking the minimum vertex boundary size for a -set. It develops a compression framework using down- and left-compressions and proves that extremal sets can be taken as compressed and ordered by the box order, yielding explicit lower bounds such as those involving , attained by the prefix set . The main result (Theorem finite) expresses the minimum as with intervals and boundary constants , and a corollary reduces the problem to ; both rely on reductions to compressed sets and a five-case analysis. The work discusses limitations with tensor products due to disconnection and outlines extensions to higher dimensions as future directions.

Abstract

For a graph and a nonempty set , the \emph{vertex boundary} of , denoted by , is defined to be the set of vertices that are not in but have at least one neighbor in . In this paper, for being a strong product of two paths, we determine the cases in which is minimized.

Paper Structure

This paper contains 4 sections, 10 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proofs of Lemma \ref{['2d']} and Lemma \ref{['2din']}
  3. Proof of Theorem \ref{['finite']}
  4. Remarks

Key Result

Theorem 1.1

Let $S$ be a set of $k$ vertices in $\square_{i=1}^n P_{\mathbb N_0}$. Then $|\partial_{\square_{i=1}^n P_{\mathbb N_0}}(S)|\ge |\partial_{\square_{i=1}^n P_{\mathbb N_0}}(S_0)|$, where $S_0$ is the set of the first $k$ vertices in the simplicial order.

Figures (1)

  • Figure 1: $k=4$, $k=13$, and $k=23$.

Theorems & Definitions (20)

  • Theorem 1.1: Wang and Wang WW, Bollobás and Leader BL2
  • Theorem 1.2: Bollobás and Leader BL2
  • Theorem 1.3: Veomett and Radcliffe VR
  • Theorem 1.4
  • Example 1.5
  • Lemma 1.7
  • Lemma 1.8
  • Lemma 2.1
  • proof
  • proof : Proof of Lemma \ref{['2d']}
  • ...and 10 more