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On Hamiltonian paths in distance graphs. (English)
Appl. Math. Lett. 24, No. 7, 1075-1079 (2011).
Summary: For a finite set $D \subseteq \Bbb N$ with gcd$(D) = 1$, we prove the existence of some $n \in \Bbb N$ such that the distance graph $P^D_n$ with vertex set $\{0,1,\cdots ,n - 1\}$ in which two vertices $u$ and $v$ are adjacent exactly if $|u - v| \in D$, has a Hamiltonian path with endvertices 0 and $n - 1$. This settles a conjecture posed by {\it L. D. Penso, D. Rautenbach} and {\it J. L. Szwarcfiter} [“Long cycles and paths in distance graphs,” Discrete Math. 310, No. 23, 3417‒3420 (2010; Zbl 1221.05221)].