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Discrete Mathematics and Combinatorics

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Articles 61 - 73 of 73

Full-Text Articles in Geometry and Topology

Conical Existence Of Closed Curves On Convex Polyhedra, Joseph O'Rourke, Costin Vîlcu Feb 2011

Conical Existence Of Closed Curves On Convex Polyhedra, Joseph O'Rourke, Costin Vîlcu

Computer Science: Faculty Publications

Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support …

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Interval Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy Jan 2011

Interval Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the notion of interval semirings are introduced. The authors study and analyse semirings algebraically. Methods are given for the construction of non-associative semirings using loops and interval semirings or interval loops and semirings. Another type of non-associative semirings are introduced using groupoids and interval semirings or interval groupoids and semirings. Examples using integers and modulo integers are given. Also infinite semirings which are semifields are given using interval semigroups and semirings or semigroups and interval semirings or using groups and interval semirings. Interval groups are introduced to construct interval group interval semirings, and properties related with them …

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Interval Semigroups, Florentin Smarandache, W.B. Vasantha Kandasamy Jan 2011

Interval Semigroups, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce the notion of interval semigroups using intervals of the form [0, a], a is real. Several types of interval semigroups like fuzzy interval semigroups, interval symmetric semigroups, special symmetric interval semigroups, interval matrix semigroups and interval polynomial semigroups are defined and discussed. This book has eight chapters. The main feature of this book is that we suggest 241 problems in the eighth chapter. In this book the authors have defined 29 new concepts and illustrates them with 231 examples. Certainly this will find several applications. The authors deeply acknowledge Dr. Kandasamy for the proof reading …

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Slider-Pinning Rigidity: A Maxwell-Laman-Type Theorem, Ileana Streinu, Louis Theran Dec 2010

Slider-Pinning Rigidity: A Maxwell-Laman-Type Theorem, Ileana Streinu, Louis Theran

Computer Science: Faculty Publications

We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley’s direction networks.

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Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke May 2009

Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke

Computer Science: Faculty Publications

The Yao graph for k = 4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners.

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Intrinsic Linking And Knotting Are Arbitrarily Complex, Erica Flapan, Blake Mellor, Ramin Naimi Jan 2008

Intrinsic Linking And Knotting Are Arbitrarily Complex, Erica Flapan, Blake Mellor, Ramin Naimi

Mathematics, Statistics and Data Science Faculty Works

We show that, given any n and α, every embedding of any sufficiently large complete graph in R3 contains an oriented link with components Q1, ..., Qn such that for every i≠j, $|\lk(Q_i,Q_j)|\geq\alpha$ and |a2(Qi)|≥α, where a2(Qi) denotes the second coefficient of the Conway polynomial of Qi.

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Weight Systems For Milnor Invariants, Blake Mellor Jan 2008

Weight Systems For Milnor Invariants, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram.

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Chord Diagrams And Gauss Codes For Graphs, Thomas Fleming, Blake Mellor Jan 2006

Chord Diagrams And Gauss Codes For Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is …

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Tree Diagrams For String Links, Blake Mellor Jan 2006

Tree Diagrams For String Links, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

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A Constructive Proof Of Ky Fan's Generalization Of Tucker's Lemma, Timothy Prescott '02, Francis E. Su Aug 2005

A Constructive Proof Of Ky Fan's Generalization Of Tucker's Lemma, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sn that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.

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Lower Bounds For Simplicial Covers And Triangulations Of Cubes, Adam Bliss '03, Francis E. Su Apr 2005

Lower Bounds For Simplicial Covers And Triangulations Of Cubes, Adam Bliss '03, Francis E. Su

All HMC Faculty Publications and Research

We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their …

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Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor Jan 2004

Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

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Applications Of Graph Theory To Separability, Stephen Young Jan 2002

Applications Of Graph Theory To Separability, Stephen Young

Mathematical Sciences Technical Reports (MSTR)

Let S be a surface with a triangular tiling T. Let R be a reflection a side of one of the triangles; so that R is an orientation reversing isometry of the surface. Define M = {s in S |S : Rs = s}. We then say that the surface S separates along the reflection R if S-R has two components. This paper considers the applications of graph theoretic methods to determining whether a reflection is separating or not and compares the algorithmic efficiency of these methods to the current known methods.

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