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The edge-isoperimetric inequality for powers of cycles

Kristiyan Vasilev

Abstract

This note provides a complete solution to a certain version of the edge-isoperimetric problem for powers of a cycle graph. Namely, it shows that the maximum number of edges inside a vertex subset of $C_n^s$ of size $k$ is achieved by a set of $k$ consecutive vertices.

The edge-isoperimetric inequality for powers of cycles

Abstract

This note provides a complete solution to a certain version of the edge-isoperimetric problem for powers of a cycle graph. Namely, it shows that the maximum number of edges inside a vertex subset of of size is achieved by a set of consecutive vertices.
Paper Structure (9 sections, 10 theorems, 20 equations, 1 table)

This paper contains 9 sections, 10 theorems, 20 equations, 1 table.

Table of Contents

  1. MSC 2020:
  2. Keywords:
  3. Introduction
  4. Notation.
  5. Application of general bounds
  6. Bound from Turán's theorem
  7. Spectral bound
  8. Numerical comparison
  9. Proof of the main result

Key Result

Theorem 1

If $n$, $k$, and $s$ are positive integers such that $n\geq k$ and $n > s$, then the maximum is attained by any set of $k$ consecutive vertices of $C_n^s$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2: Turán's
  • Lemma 1
  • proof
  • Corollary 1
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 7 more