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Triangulations of 3-dimensional polyhedron are partitions of the polyhedron with tetrahedra in a face-to-face fashion without introducing new vertices. Schönhardt (Math. Ann. 89:309–312, 1927), Bagemihl (Amer. Math. Mon. 55:411–413, 1948), Kuperberg (Personal communication 2011) and others constructed special polyhedra in such a way that clever one line geometric reasons imply nontriangulability. Rambau (Comb. Comput. Geom. 52:501–516, 2005) proved that twisted prisms over n-gons are nontriangulable. Our approach for proving polyhedra are nontriangulable is to show that partitions with tetrahedra, which we call tilings, do not exist even if the face-to-face-restriction is relaxed. First we construct a polyhedron which is tileable but is not triangulable. Then we revisit Rambau type twisted prisms. In fact we consider a slightly different class of polyhedra, and prove that these new twisted prisms are nontileable, thus are nontriangulable. We also show that one can twist the regular dodecahedron so that it becomes nontileable, which is abstracted to a new family of nontileable polyhedra, called nonconvex twisted pentaprisms. © 2015, The Managing Editors.
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A Skolem sequence can be thought of as a labelled path where two vertices with the samelabelarethatdistanceapart.Thisconcepthasnaturallybeengeneralizedto labellingsofothergraphs,butalwaysusingatmosttwoofanyintegerlabel.Giventhat more than two vertices can be mutually distance d apart, wedefine a new generalization of aSkolemsequenceongraphsthatwecallaproperSkolemlabelling.Thisbrings rise tothequestion;use to proper Skolemlabel a graph?"This will be known as the Skolem number of the graph. In this paper we give the Skolem number for cycles and grid graphs,while also providing other related results along the way. Published by Digital Commons@Georgia Southern, 2021. © 2021 Georgia Southern University. All rights reserved.
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A proper Skolem labelling of a graph G is a function assigning a positive integer to each vertex of G such that any two vertices assigned the same integer are that distance apart in the graph. The Skolem number of a graph is smallest number n such that there exists a proper Skolem labelling only using the positive integers less than or equal to n. In this paper, we will begin by proving the Skolem number for another family of subgraphs of the hexagonal lattice and then prove the Skolem number for two families of subgraphs of the Kagome Lattice. © 2025 Georgia Southern University. All rights reserved.
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We introduce a variant of the Kronecker product, called the regional Kronecker product, that can be used to build new, larger multiple-pair latin squares from existing multiple-pair latin squares. We present applications to the existence and orthogonality of multiple-pair latin squares. © 2019 Elsevier B.V.
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An (a, b)-Sudoku pair Latin square is a Latin square that is simult-aneously an (a, b)-Sudoku Latin square and a (b, a)-Sudoku Latin square. While (a, b)-Sudoku Latin squares are known to exist for any positive integers a and b, the pairs a, b for which an (a, b)-Sudoku pair Latin square exists are largely unknown. In this article we establish the existence of (a, b)-Sudoku pair Latin squares for an infinite collection of pairs (a, b). Our results show that a (3, b)-Sudoku pair Latin square can be constructed for any positive integer b. ©The author(s).
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A rokudoku-pair square is an order-6 sudoku Latin square for both 2 x 3 and 3 x 2 tiling regions simultaneously. We count the distinct rokudoku-pair squares as well as orbits under the action of a suitable group. Our arguments employ group actions and list colorings of graphs. As an application we determine which rokudoku-pair squares are based on groups. © 2021. All Rights Reserved.
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