Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Perfect codes in circulant graphs of degree $p^l-1$

Xiaomeng Wang, Oriol Serra, Shou-Jun Xu, Sanming Zhou

Abstract

A perfect code in a graph is an independent set of the graph such that every vertex outside the set is adjacent to exactly one vertex in the set. A circulant graph is a Cayley graph of a cyclic group. In this paper we study perfect codes in circulant graphs of degree $p^l - 1$, where $p$ is a prime and $l \ge 1$. We obtain a necessary and sufficient condition for such a circulant graph to admit perfect codes, give a construction of all such circulant graphs which admit perfect codes, and prove a lower bound on the number of distinct perfect codes in such a circulant graph. This extends known results for the case $l=1$ and provides insight on the general problem on the existence and structure of perfect codes in circulant graphs.

Perfect codes in circulant graphs of degree $p^l-1$

Abstract

A perfect code in a graph is an independent set of the graph such that every vertex outside the set is adjacent to exactly one vertex in the set. A circulant graph is a Cayley graph of a cyclic group. In this paper we study perfect codes in circulant graphs of degree , where is a prime and . We obtain a necessary and sufficient condition for such a circulant graph to admit perfect codes, give a construction of all such circulant graphs which admit perfect codes, and prove a lower bound on the number of distinct perfect codes in such a circulant graph. This extends known results for the case and provides insight on the general problem on the existence and structure of perfect codes in circulant graphs.
Paper Structure (11 sections, 24 theorems, 42 equations, 1 figure)

This paper contains 11 sections, 24 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some history
  3. Periodic sets
  4. Main results
  5. Organisation of the paper
  6. Preliminaries
  7. Proof of Theorem \ref{['th:thplp']}
  8. Proof of Theorem \ref{['th:th2.2.1']}
  9. Proof of Theorem \ref{['cor:cor2.2']}
  10. Final remarks
  11. A formula

Key Result

Theorem 1.2

Let $n \ge 3$ be an integer, and let $\mathrm{Cay}(\mathbb{Z}_n, S)$ be a connected non-complete circulant graph with order $n$ and degree $p^{\ell}-1$, where $p$ is a prime and $\ell \ge 1$ is an integer. Then $\mathrm{Cay}(\mathbb{Z}_n, S)$ admits a perfect code if and only if $p^{\ell}$ divides $

Figures (1)

  • Figure 1: In $G=\mathbb{Z}_{90}$, $S_0=\{0,1,15,16,31,59,74,75,89\}$ is a pyramidal set with the longest admissible subgroup series $H_1=\langle 45\rangle, H_2=\langle 15\rangle, H_3=\langle 3 \rangle$. For $i = 1, 2, 3$, the members of $G/H_i$ in red colour form $S_0/H_i$. This diagram looks like a pyramid, hence the eponym.

Theorems & Definitions (43)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 33 more