Hamiltonian cycles in graphs were first studied in the 1850s. Since then, an impressive amount of research has been dedicated to identifying classes of graphs that allow Hamiltonian cycles, and to related questions. The corresponding decision problem, that asks whether a given graph is Hamiltonian (i.\,e.\ admits a Hamiltonian cycle), is one of Karp's famous NP-complete problems. In this paper we study graphs of bounded degree that are \emph{far} from being Hamiltonian, where a graph $G$ on $n$ vertices is \emph{far} from being Hamiltonian, if modifying a constant fraction of $n$ edges is necessary to make $G$ Hamiltonian. We give an explicit deterministic construction of a class of graphs of bounded degree that are locally Hamiltonian, but (globally) far from being Hamiltonian. Here, \emph{locally Hamiltonian} means that every subgraph induced by the neighbourhood of a small vertex set appears in some Hamiltonian graph. More precisely, we obtain graphs which differ in $\Theta(n)$ edges from any Hamiltonian graph, but non-Hamiltonicity cannot be detected in the neighbourhood of $o(n)$ vertices. Our class of graphs yields a class of hard instances for one-sided error property testers with linear query complexity. It is known that any property tester (even with two-sided error) requires a linear number of queries to test Hamiltonicity (Yoshida, Ito, 2010). This is proved via a randomised construction of hard instances. In contrast, our construction is deterministic. So far only very few deterministic constructions of hard instances for property testing are known. We believe that our construction may lead to future insights in graph theory and towards a characterisation of the properties hat are testable in the bounded-degree model.

翻译：1850年代首次研究了汉密尔顿周期的图表。 自那时以来, 大量的研究都用于确定允许汉密尔顿周期的图表类别, 以及相关问题。 相应的决定问题, 询问某个特定图表是否为汉密尔顿( i.\, e.\ 承认汉密尔顿周期 ), 是Karp 著名的 NP 完整的问题之一 。 在本文中, 我们从汉密尔顿州开始研究约束度的图表, 其范围不是 汉密尔顿州。 自那时以来, 大量的研究都致力于确定允许汉密尔顿州循环周期的图表类别类别。 如果需要修改一个固定部分的美尔密尔密尔顿周期( e., e., e. e. e. i.) 明确确定某类封闭度的图表类型, 但它的构造可能是已知的。 在汉密尔密尔顿州( r. i. i. i. i.) 中, 任何已知的直径的直线值, 直径直径直径直径, 直径直径直径直到直径直径直径直径直到直到直至直至直至直至直径直至直至直至直至直。 。 。 。 。 。 。 。 。 。 。 。 。 。 我们的根根根根根根根根根根直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至直至