On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs

Christian Löwenstein,Roman Soták,Dieter Rautenbach

doi:10.46298/dmtcs.1261

Abstract

Graph Theory For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set &#x007b; 0,1,\textellipsis,n-1&#x007d; and two vertices i and j of GnD are adjacent exactly if |j-i|∈D. The condition gcd(D)=1 is necessary for a distance graph GnD being connected. Let D=&#x007b;d1,d2&#x007d;⊆ℕ be such that d1>d2 and gcd(d1,d2)=1. We prove the following results. If n is sufficiently large in terms of D, then GnD has a Hamiltonian path with endvertices 0 and n-1. If d1d2 is odd, n is even and sufficiently large in terms of D, then GnD has a Hamiltonian cycle. If d1d2 is even and n is sufficiently large in terms of D, then GnD has a Hamiltonian cycle.

Highlights

For a finite set of positive integers D ⊆ N, the infinite distance graph GD has vertex set V (GD) = Z and two vertices u and v of GD are adjacent exactly if |u − v| ∈ D
For a graph G and a subset U ⊆ V (G) of the vertex set, we denote by G[U ] the subgraph of G induced by U
For a positive integer n ∈ N, the distance graph GDn = GD[[0, n − 1]] is the subgraph of GD induced by the vertices in [0, n − 1]

Summary

Introduction

For a finite set of positive integers D ⊆ N, the infinite distance graph GD has vertex set V (GD) = Z and two vertices u and v of GD are adjacent exactly if |u − v| ∈ D. Distance graphs generalize the very well-studied class of circulant graphs [2, 16, 17, 26]. In [25, 27, 15] the existence of long paths and cycles in distance graphs is studied. Theorem 1 (Lowenstein et al [21]) For a finite set D ⊆ N with |D| ≥ 2 and gcd(D) = 1, there are infinitely many n ∈ N such that GDn has a Hamiltonian cycle and GDn+1 has a Hamiltonian path with endvertices 0 and n.

The structure of GD

Hamiltonian paths of GDn

Hamiltonian cycles of GDn