On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs

Christian Löwenstein (1) , Dieter Rautenbach (1) , Roman Soták (2)

1 Institut für Optimierung und Operations Research
2 Institute of Mathematics [Kosice, Slovakia]

For a positive integer n∈ℕ and a set D⊆ ℕ, the distance graph GnD has vertex set { 0,1,\textellipsis,n-1} 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={d1,d2}⊆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.

Mots clés: Distance graph; circulant graph; Hamiltonian path; Hamiltonian cycle

Domaines: Mathématique discrète [cs.DM]

Christian Löwenstein, Dieter Rautenbach, Roman Soták. On Hamiltonian Paths and Cycles in Sufficiently Large Distance Graphs. Discrete Mathematics and Theoretical Computer Science, 2014, Vol. 16 no. 1 (1), pp.7--30. ⟨10.46298/dmtcs.1261⟩. ⟨hal-01179214⟩