On graphs of bounded degree that are far from being Hamiltonian

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. It remains NP-complete on planar cubic graphs. In this paper we study graphs of bounded degree that are far from being Hamiltonian, where a graph G on n vertices is far from being Hamiltonian, if modifying a constant fraction of n edges is necessary to make G Hamiltonian. We exhibit classes of graphs of bounded degree that are locally Hamiltonian, i.e. every subgraph induced by the neighbourhood of a small vertex set appears in some Hamiltonian graph, but that are far from being Hamiltonian. We then use these classes to obtain a lower bound in property testing. We show that in the bounded-degree graph model, Hamiltonicity is not testable with one-sided error probability and query complexity o(n). This contrasts the known fact that on planar (or minor-free) graph classes, Hamiltonicity is testable with constant query complexity in the bounded-degree graph model with two-sided error. Our proof is an intricate construction that shows how to turn a d-regular graph into a graph that is far from being Hamiltonian, and we use d-regular expander graphs to maintain local Hamiltonicity.