A Computer-Aided Decomposition Of The Complete Digraph Into Orientations Of K4 − E With A Double Edge, 2018 Illinois Mathematics and Science Academy

#### A Computer-Aided Decomposition Of The Complete Digraph Into Orientations Of K4 − E With A Double Edge, Hanson Hao

*The International Student Science Fair 2018*

The abstract is available as an Additional File.

A Computer-Aided Decomposition Of The Complete Digraph Into Orientations Of K4 − E With A Double Edge, 2018 Illinois Mathematics and Science Academy

#### A Computer-Aided Decomposition Of The Complete Digraph Into Orientations Of K4 − E With A Double Edge, Hanson Hao '19, Claudia Zhu '18

*Student Publications & Research*

The abstract is available as an Additional File.

Partition Problems In High Dimensional Boxes, 2018 ETH Zurich

#### Partition Problems In High Dimensional Boxes, Matija Bucic, Bernard Lidicky, Jason Long, Adam Zsolt Wagner

*Mathematics Publications*

Alon, Bohman, Holzman and Kleitman proved that any partition of a d-dimensional discrete box into proper sub-boxes must consist of at least 2d sub-boxes. Recently, Leader, Milicevic and Tan considered the question of how many odd-sized proper boxes are needed to partition a d-dimensional box of odd size, and they asked whether the trivial construction consisting of 3d boxes is best possible. We show that approximately 2.93d boxes are enough, and consider some natural generalisations.

Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, 2018 Louisiana State University and Agricultural and Mechanical College

#### Dimers On Cylinders Over Dynkin Diagrams And Cluster Algebras, Maitreyee Chandramohan Kulkarni

*LSU Doctoral Dissertations*

This dissertation describes a general setting for dimer models on cylinders over Dynkin diagrams which in type A reduces to the well-studied case of dimer models on a disc. We prove that all Berenstein--Fomin--Zelevinsky quivers for Schubert cells in a symmetric Kac--Moody algebra give rise to dimer models on the cylinder over the corresponding Dynkin diagram. We also give an independent proof of a result of Buan, Iyama, Reiten and Smith that the corresponding superpotentials are rigid using the dimer model structure of the quivers.

Facial Unique-Maximum Colorings Of Plane Graphs With Restriction On Big Vertices, 2018 Iowa State University

#### Facial Unique-Maximum Colorings Of Plane Graphs With Restriction On Big Vertices, Bernard Lidicky, Kacy Messerschmidt, Riste Škrekovski

*Mathematics Publications*

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This ...

Templates For Representable Matroids, 2018 Louisiana State University and Agricultural and Mechanical College

#### Templates For Representable Matroids, Kevin Manuel Grace

*LSU Doctoral Dissertations*

The matroid structure theory of Geelen, Gerards, and Whittle has led to a hypothesis that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a *perturbation*) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a *template* to describe these perturbations in more detail. In this dissertation, we determine these templates for various classes and use them to prove results about representability, extremal functions, and excluded minors.

Chapter 1 gives a brief introduction to ...

Average Mixing Matrix Of Trees, 2018 University of Waterloo

#### Average Mixing Matrix Of Trees, Chris Godsil, Krystal Guo, John Sinkovic

*Electronic Journal of Linear Algebra*

The rank of the average mixing matrix of trees with all eigenvalues distinct, is investigated. The rank of the average mixing matrix of a tree on n vertices with n distinct eigenvalues is bounded above by ⌈n/2⌉. Computations on trees up to 20 vertices suggest that the rank attains this upper bound most of the time. An infinite family of trees whose average mixing matrices have ranks which are bounded away from this upper bound, is given. A lower bound on the rank of the average mixing matrix of a tree, is also given.

Two Results In Drawing Graphs On Surfaces, 2018 Louisiana State University and Agricultural and Mechanical College

#### Two Results In Drawing Graphs On Surfaces, Joshua E. Fallon

*LSU Doctoral Dissertations*

In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian ...

$\Gamma'$-Realizability And Other Musings On Inverse Domination, 2018 Dalton State College

#### $\Gamma'$-Realizability And Other Musings On Inverse Domination, John Asplund, Joe Chaffee, James M. Hammer Iii, Matt Noble

*Theory and Applications of Graphs*

We introduce and study $\gamma'$-realizable sequences. For a finite, simple graph $G$ containing no isolated vertices, $I \subseteq V(G)$ is said to be an \emph{inverse dominating set} if $I$ dominates all of $G$ and $I$ is contained by the complement of some minimum dominating set $D$. Define a sequence of positive integers $(x_1, \ldots , x_n)$ to be \emph{$\gamma'$-realizable} if there exists a graph $G$ having exactly $n$ distinct minimum dominating sets $D_1, \ldots, D_n$ where for each $i \in \{1, \ldots, n\}$, the minimum size of an inverse dominating set in $V(G) \setminus D_i ...

The Search For The Cyclic Sieving Phenomenon In Plane Partitions, 2018 DePaul University

#### The Search For The Cyclic Sieving Phenomenon In Plane Partitions, William J. Asztalos

*DePaul Discoveries*

The efforts of this research project are best understood in the context of the subfield of dynamical combinatorics, in which one enumerates a set of combinatorial objects by defining some action to guide the search for underlying structures. While there are many examples with varying degrees of complexity, the necklace problem, which concerns the possible unique configurations of beads in a ring up to rotational symmetry, is a well-known example. Though this sort of approach to enumeration has been around for a century or more, activity in this area has intensified in the last couple of decades. Perhaps the most ...

Additive List Coloring Of Planar Graphs With Given Girth, 2018 Northern Kentucky University

#### Additive List Coloring Of Planar Graphs With Given Girth, Axel Brandt, Sogol Jahanbekam, Jennifer Diemunsch

*Faculty Publications*

An additive coloring of a graph G is a labeling of the vertices of G from {1,2,...,k} such that two adjacent vertices have distinct sums of labels on their neighbors. The least integer k for which a graph G has an additive coloring is called the additive coloring number of G, denoted \luck(G). Additive coloring is also studied under the names lucky labeling and open distinguishing. In this paper, we improve the current bounds on the additive coloring number for particular classes of graphs by proving results for a list version of additive coloring. We apply the ...

On The Distance And Distance Signless Laplacian Eigenvalues Of Graphs And The Smallest Gersgorin Disc, 2018 Indian Institute of Technology Kharagpur

#### On The Distance And Distance Signless Laplacian Eigenvalues Of Graphs And The Smallest Gersgorin Disc, Fouzul Atik, Pratima Panigrahi

*Electronic Journal of Linear Algebra*

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless ...

Golden Arm: A Probabilistic Study Of Dice Control In Craps, 2018 Monmouth University

#### Golden Arm: A Probabilistic Study Of Dice Control In Craps, Donald R. Smith, Robert Scott Iii

*UNLV Gaming Research & Review Journal*

This paper calculates how much control a craps shooter must possess on dice outcomes to eliminate the house advantage. A golden arm is someone who has dice control (or a rhythm roller or dice influencer). There are various strategies for dice control in craps. We discuss several possibilities of dice control that would result in several different mathematical models of control. We do not assert whether dice control is possible or not (there is a lack of published evidence). However, after studying casino-legal methods described by dice-control advocates, we can see only one realistic mathematical model that describes the resulting ...

Secure Multiparty Protocol For Differentially-Private Data Release, 2018 Boise State University

#### Secure Multiparty Protocol For Differentially-Private Data Release, Anthony Harris

*Boise State University Theses and Dissertations*

In the era where big data is the new norm, a higher emphasis has been placed on models which guarantees the release and exchange of data. The need for privacy-preserving data arose as more sophisticated data-mining techniques led to breaches of sensitive information. In this thesis, we present a secure multiparty protocol for the purpose of integrating multiple datasets simultaneously such that the contents of each dataset is not revealed to any of the data owners, and the contents of the integrated data do not compromise individual’s privacy. We utilize privacy by simulation to prove that the protocol is ...

The Average Measure Of A K-Dimensional Simplex In An N-Cube, 2018 Missouri State University

#### The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter

*MSU Graduate Theses*

Within an n-dimensional unit cube, a number of k-dimensional simplices can be formed whose vertices are the vertices of the n-cube. In this thesis, we analyze the average measure of a k-simplex in the n-cube. We develop exact equations for the average measure when k = 1, 2, and 3. Then we generate data for these cases and conjecture that their averages appear to approach n^{k/2} times some constant. Using the convergence of Bernstein polynomials and a k-simplex Bernstein generalization, we prove the conjecture is true for the 1-simplex and 2-simplex cases. We then develop a generalized formula for ...

On Coding For Partial Erasure Channels, 2018 University of Nebraska - Lincoln

#### On Coding For Partial Erasure Channels, Carolyn Mayer

*Dissertations, Theses, and Student Research Papers in Mathematics*

Error correcting codes have been essential to the technology we use in everyday life in digital storage, wireless communication, barcodes, and much more. Different channel models are used for different types of communication (for example, if information is sent to one person or to many people) and different types of errors. Partial erasure channels were recently introduced to model applications in which some information remains after an erasure event. These remnants of information may be used to increase the chances of successful decoding. We introduce a new partial erasure channel in which partial erasures correspond to individual bit erasures in ...

Covering Arrays For Equivalence Classes Of Words, 2018 East Tennessee State University

#### Covering Arrays For Equivalence Classes Of Words, Joshua Cassels, Anant Godbole

*Undergraduate Honors Theses*

Covering arrays for words of length t over a d letter alphabet are k × n arrays with entries from the alphabet so that for each choice of t columns, each of the d^{t} t-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case, words are equivalent if they induce the same partition of a t element set. In the second case, words of the same weighted sum are equivalent. In both cases we produce logarithmic upper bounds on ...

A New Upper Bound For The Diameter Of The Cayley Graph Of A Symmetric Group, 2018 College of William and Mary

#### A New Upper Bound For The Diameter Of The Cayley Graph Of A Symmetric Group, Hangwei Zhuang

*Undergraduate Honors Theses*

Given a finite symmetric group S_n and a set S of generators, we can represent the group as a Cayley graph. The diameter of the Cayley graph is the largest distance from the identity to any other elements. We work on the conjecture that the diameter of the Cayley graph of a finite symmetric group S_n with S ={(12),(12...n)} is at most $ C(n,2). Our main result is to show that the diameter of the graph of S_n is at most (3n^2-4n)/2.

Graphs With Few Spanning Substructures, 2018 University of Nebraska - Lincoln

#### Graphs With Few Spanning Substructures, Jessica De Silva

*Dissertations, Theses, and Student Research Papers in Mathematics*

In this thesis, we investigate a number of problems related to spanning substructures of graphs. The first few chapters consider extremal problems related to the number of forest-like structures of a graph. We prove that one can find a threshold graph which contains the minimum number of spanning pseudoforests, as well as rooted spanning forests, amongst all graphs on n vertices and e edges. This has left the open question of exactly which threshold graphs have the minimum number of these spanning substructures. We make progress towards this question in particular cases of spanning pseudoforests.

The final chapter takes on ...

Vector Partitions, 2018 East Tennessee State University

#### Vector Partitions, Jennifer French

*Electronic Theses and Dissertations*

Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The ...