Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$s$-Stable Kneser Graph are Hamiltonian

Agustina V. Ledezma, Adrián G. Pastine

Abstract

The Kneser Graph $K(n,k)$ has as vertices all $k$-subsets of $\{1,\ldots,n\}$ and edges connecting two vertices if they are disjoint. The $s$-stable Kneser Graph $K_{s-stab}(n, k)$ is obtained from the Kneser graph by deleting vertices with elements at cyclic distance less than $s$. In this article we show that connected $s$-Stable Kneser graph are Hamiltonian.

$s$-Stable Kneser Graph are Hamiltonian

Abstract

The Kneser Graph has as vertices all -subsets of and edges connecting two vertices if they are disjoint. The -stable Kneser Graph is obtained from the Kneser graph by deleting vertices with elements at cyclic distance less than . In this article we show that connected -Stable Kneser graph are Hamiltonian.
Paper Structure (4 sections, 3 theorems, 2 equations, 3 figures, 1 table)

This paper contains 4 sections, 3 theorems, 2 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Class Graph and proof of the main theorem
  4. Acknowledgements

Key Result

Theorem 1

Let $k\geq 1$ and $s\geq 3$. The graph $K_{s-stab}(n, k)$ is Hamiltonian if and only if $n\geq sk$. The graph $K_{2-stab}(n, k)$ is Hamiltonian if and only if $n\geq 2k+1$.

Figures (3)

  • Figure 1: Grid of the vertex $\lbrace 1,3,5 \rbrace$ in $K_{2-stab}(9, 3)$.
  • Figure 2: Grid of vertices in $K_{2-stab}(9, 3)$.
  • Figure :

Theorems & Definitions (16)

  • Theorem 1
  • Definition 1
  • Claim 1
  • Claim 2
  • Lemma 1
  • proof
  • Definition 2
  • Claim 3
  • proof
  • Claim 4
  • ...and 6 more