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Towards a Complete Local-Global Principle

Nikola Kuzmanovski

Abstract

Ahlswede and Cai proved that if a simple graph has nested solutions (NS) under the edge-isoperimetric problems, and the lexicographic (lex) order produces NS for its second cartesian power,then the lex order produces NS for any finite cartesian power. Under very general assumptions, we prove that if a graph and its second cartesian power have NS,then so does any finite cartesian power. Harper asked if this is true without any restriction. We also conjecture that it is. All graphs studied in the literature for which the lex order is optimal are regular. This lead Bezrukov and Elsässer to conjecture that if the lex order is optimal for the second cartesian power, then the original graph is regular. A counterexample to this conjecture is provided.

Towards a Complete Local-Global Principle

Abstract

Ahlswede and Cai proved that if a simple graph has nested solutions (NS) under the edge-isoperimetric problems, and the lexicographic (lex) order produces NS for its second cartesian power,then the lex order produces NS for any finite cartesian power. Under very general assumptions, we prove that if a graph and its second cartesian power have NS,then so does any finite cartesian power. Harper asked if this is true without any restriction. We also conjecture that it is. All graphs studied in the literature for which the lex order is optimal are regular. This lead Bezrukov and Elsässer to conjecture that if the lex order is optimal for the second cartesian power, then the original graph is regular. A counterexample to this conjecture is provided.
Paper Structure (4 sections, 12 theorems, 13 equations)

This paper contains 4 sections, 12 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Uniqueness of $\mathcal{L}$ and $\mathcal{C}$ in $G^2$
  3. Towards a Complete Local-Global Principle
  4. Acknowledgments

Key Result

Lemma 1.1

If $G=(V,E)$ is a regular graph with degree $n$ and $A\subseteq V$ then

Theorems & Definitions (20)

  • Lemma 1.1
  • Theorem 1.2: Ahlswede-Cai AhlswedeCai1997_II, 1997
  • Lemma 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Lemma 2.1
  • Lemma 2.2: Bezrukov Bezrukov1999Equiv ,1999
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • ...and 10 more