Abstract
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.
Highlights
A Hamiltonian cycle in a graph G is a cycle that visits every vertex of G exactly once
A graph G is Hamiltonian if G contains a Hamiltonian cycle
We show that any one-sided error property tester with sublinear query complexity needs to accept any locally Hamiltonian graph
Summary
A Hamiltonian cycle in a graph G is a cycle that visits every vertex of G exactly once. It is known that some properties are constant-query testable, including subgraph-freeness, kedge connectivity, cycle-freeness, being Eulerian, degree-regularity Goldreich and Ron (2002), minorfreeness Benjamini et al (2010); Hassidim et al (2009); Kumar et al (2019), hyperfinite properties Newman and Sohler (2013), k-vertex connectivity Yoshida and Ito (2012); Forster et al (2020), and. Since every graph in our constructed class is locally Hamiltonian and far from being Hamiltonian we get the following, previously known lower-bound (see Yoshida and Ito (2010); Goldreich (2020)) as a direct consequence of Theorem 5.5 (cf Corollary 6.2). Hamiltonicity is not testable with one-sided error and query complexity o(n) in the bounded-degree model This provides evidence that using deterministic constructions is a viable route for finding lower bounds for property testing.
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