  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 that are testable in the bounded-degree
model.
