doi:10.1016/j.comgeo.2017.05.009engClaverol, MercèGarcía, AlfredoGarijo, DeliaSeara, CarlosTejel, JavierOn Hamiltonian alternating cycles and pathsART-2018-101839We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. For point sets in general position, our main result is that it is always possible to obtain a 1-plane Hamiltonian alternating cycle. When the point set is in convex position, we prove that every Hamiltonian alternating cycle with minimum number of crossings is 1-plane, and provide O(n) and O(n2) time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.2018http://zaguan.unizar.es/record/7810110.1016/j.comgeo.2017.05.009http://zaguan.unizar.es/record/78101oai:zaguan.unizar.es:78101info:eu-repo/grantAgreement/ES/DGA/E58info:eu-repo/grantAgreement/ES/MINECO-FEDER/MTM2014-60127-Pinfo:eu-repo/grantAgreement/ES/MINECO-FEDER/MTM2015-63791-RCOMPUTATIONAL GEOMETRY-THEORY AND APPLICATIONS 68 (2018), 146-166by-nc-ndhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccess