Adler, I orcid.org/0000-0002-9667-9841 and Köhler, N (2022) An explicit construction of graphs of bounded degree that are far from being Hamiltonian. Discrete Mathematics and Theoretical Computer Science, 24 (1). 3. ISSN 1462-7264
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 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 give an explicit deterministic construction of a class of graphs of bounded degree that are locally Hamiltonian, but (globally) far from being Hamiltonian. Here, 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2022 by the author(s). Distributed under a Creative Commons Attribution 4.0 International License. |
Keywords: | Hamiltonian cycle, property testing, bounded-degree graphs, bounded-degree model, lower bound |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 31 Jan 2022 15:45 |
Last Modified: | 16 Mar 2023 14:21 |
Status: | Published |
Publisher: | Episciences |
Identification Number: | 10.46298/dmtcs.7109 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:182980 |