dc.contributor.author Bose, Prosenjit dc.contributor.author Cano Vila, María del Pilar dc.contributor.author Saumell Mendiola, Maria dc.contributor.author Silveira, Rodrigo Ignacio dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtiques dc.date.accessioned 2020-12-21T10:00:49Z dc.date.available 2021-08-01T00:33:51Z dc.date.issued 2020-08 dc.identifier.citation Bose, P. [et al.]. Hamiltonicity for convex shape Delaunay and Gabriel graphs. "Computational geometry: theory and applications", Agost 2020, vol. 89, p. 101629:17. dc.identifier.issn 0925-7721 dc.identifier.uri http://hdl.handle.net/2117/334699 dc.description © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ dc.description.abstract We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape $$\mathcal {C}$$ . Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k- $${DG}_{\mathcal {C}}(S)$$ , has vertex set S and edge pq provided that there exists some homothet of $$\mathcal {C}$$ with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph k- $${GG}_{\mathcal {C}}(S)$$ is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of $$\mathcal {C}$$ with p and q on its boundary. We provide upper bounds on the minimum value of k for which k- $${GG}_{\mathcal {C}}(S)$$ is Hamiltonian. Since k- $${GG}_{\mathcal {C}}(S)$$ $$\subseteq$$ k- $${DG}_{\mathcal {C}}(S)$$ , all results carry over to k- $${DG}_{\mathcal {C}}(S)$$ . In particular, we give upper bounds of 24 for every $$\mathcal {C}$$ and 15 for every point-symmetric $$\mathcal {C}$$ . We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for $$t \ge 10)$$ . These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs. dc.description.sponsorship P.B. was partially supported by NSERC. P.C. was supported by CONACyT. M.S. was supported by the Czech Science Foundation, grant number GJ19-06792Y, and by institutional support RVO:67985807. R.S. was supported by MINECO through the Ram´on y Cajal program. P.C. and R.S. were also supported by projects MINECO MTM2015-63791-R and Gen. Cat. 2017SGR1640. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922. dc.language.iso eng dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica aplicada a les ciències dc.subject.lcsh Hamiltonian systems dc.subject.other Delaunay graphs dc.subject.other Hamiltonicity dc.subject.other Gabriel graphs dc.title Hamiltonicity for convex shape Delaunay and Gabriel graphs dc.type Article dc.subject.lemac Hamilton, Sistemes de dc.contributor.group Universitat Politècnica de Catalunya. CGA - Computational Geometry and Applications dc.identifier.doi 10.1016/j.comgeo.2020.101629 dc.description.peerreviewed Peer Reviewed dc.subject.ams Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems dc.relation.publisherversion https://www.sciencedirect.com/science/article/abs/pii/S0925772120300237 dc.rights.access Open Access local.identifier.drac 28657735 dc.description.version Postprint (author's final draft) dc.relation.projectid info:eu-repo/grantAgreement/EC/H2020/734922/EU/Combinatorics of Networks and Computation/CONNECT dc.relation.projectid info:eu-repo/grantAgreement/MINECO//MTM2015-63791-R/ES/GRAFOS Y GEOMETRIA: INTERACCIONES Y APLICACIONES/ local.citation.author Bose, P.; Cano, M.; Saumell, M.; Silveira, R. local.citation.publicationName Computational geometry: theory and applications local.citation.volume 89 local.citation.startingPage 101629:1 local.citation.endingPage 101629:17
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