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Classification of perfect and total perfect codes in generalized Petersen graphs

Xiaomeng Wang, Junyang Zhang

Abstract

In a graph $Γ$, a perfect code is an independent set $C$ with the property that every vertex not in $C$ is adjacent to a unique vertex in $C$, and a total perfect code is a set $C$ of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. We classify these codes for generalized Petersen graphs.

Classification of perfect and total perfect codes in generalized Petersen graphs

Abstract

In a graph , a perfect code is an independent set with the property that every vertex not in is adjacent to a unique vertex in , and a total perfect code is a set of vertices of such that every vertex of is adjacent to a unique vertex in . We classify these codes for generalized Petersen graphs.
Paper Structure (4 sections, 8 theorems, 42 equations, 4 figures)

This paper contains 4 sections, 8 theorems, 42 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Constructions
  3. Proof of Theorem \ref{['pcode']}
  4. Proof of Theorem \ref{['tpcode']}

Key Result

Theorem 1.1

Let $C$ be a subset of the vertex set of $\mathrm{GP}(n,k)$. Then $C$ is a perfect code in $\mathrm{GP}(n,k)$ if and only if $n\equiv0\pmod{4}$, $k\equiv1\pmod{2}$ and $C=\{u_{4i+j},v_{4i+j+2}\mid i\in \mathbb{Z}_n\}$ for some $j\in\{0,1,2,3\}$.

Figures (4)

  • Figure 1: $u_{\ell},u_{\ell+1},u_{\ell+2},u_{\ell+3}\in C$
  • Figure 2: $u_0,u_1\in C$
  • Figure 4: $v_{4-k},v_4\in C$
  • Figure 5: $v_{2+k},v_{2+2k}\in C$

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['pcode']}
  • Lemma 4.1
  • proof
  • ...and 4 more