episciences.org_7109_1675592856 1675592856 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Discrete Mathematics & Theoretical Computer Science 1365-8050 01 31 2022 vol. 24, no. 1 Graph Theory An explicit construction of graphs of bounded degree that are far from being Hamiltonian Isolde Adler Noleen Köhler 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. 01 31 2022 7109 https://arxiv.org/licenses/nonexclusive-distrib/1.0 arXiv:2008.05801 10.48550/arXiv.2008.05801 https://arxiv.org/abs/2008.05801v5 https://arxiv.org/abs/2008.05801v4 https://arxiv.org/abs/2008.05801v3 10.46298/dmtcs.7109 https://dmtcs.episciences.org/7109 https://dmtcs.episciences.org/8997/pdf https://dmtcs.episciences.org/8997/pdf