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.

Source : oai:HAL:hal-01179214v1

Volume: Vol. 16 no. 1 (in progress)

Section: Graph Theory

Published on: February 4, 2014

Submitted on: February 10, 2011

Keywords: Distance graph, circulant graph, Hamiltonian path, Hamiltonian cycle,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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