Discrete Mathematics and Combinatorics Commons^™

Tree Diagrams For String Links, Blake Mellor

Blake Mellor

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.

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

Blake Mellor

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).

Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn

Blake Mellor

We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.

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

Blake Mellor

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 …

Influences Of Probability Instruction On Undergraduates' Understanding Of Counting Processes, Kayla Blyman

Theses and Dissertations--Education Sciences

Historically, students in an introductory finite mathematics course at a major university in the mid-south have struggled the most with the counting and probability unit, leading instructors to question if there was a better way to help students master the material. The purpose of this study was to begin to understand connections that undergraduate finite mathematics students are making between counting and probability. By examining student performance in counting and probability, this study provides insights that inform future instruction in courses that include counting and probability. Consequently, this study lays the groundwork for future inquiries in the field of undergraduate …

Colorings Of Hamming-Distance Graphs, Isaiah H. Harney

Theses and Dissertations--Mathematics

Hamming-distance graphs arise naturally in the study of error-correcting codes and have been utilized by several authors to provide new proofs for (and in some cases improve) known bounds on the size of block codes. We study various standard graph properties of the Hamming-distance graphs with special emphasis placed on the chromatic number. A notion of robustness is defined for colorings of these graphs based on the tolerance of swapping colors along an edge without destroying the properness of the coloring, and a complete characterization of the maximally robust colorings is given for certain parameters. Additionally, explorations are made into …

An Eternal Domination Problem In Grids, William Klostermeyer, Margaret-Ellen Messinger, Alejandro Angeli Ayello

Theory and Applications of Graphs

A dynamic domination problem in graphs is considered in which an infinite sequence of attacks occur at vertices with mobile guards; the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard. Other guards are allowed to move at the same time, and before and after each attack, the vertices containing guards must form a dominating set of the graph. The minimum number of guards that can defend the graph against such an arbitrary sequence of attacks is called the m-eviction number of the graph. In this paper, the m-eviction …

Investigating Difference Equations, Dave Ruch

Discrete Mathematics

No abstract provided.

Rainbow Turán Problems For Paths And Forests Of Stars, Daniel Johnston, Cory Palmer, Amites Sarkar

Mathematics Faculty Publications

For a fixed graph F, we would like to determine the maximum number of edges in a properly edge-colored graph on n vertices which does not contain a rainbow copy of F, that is, a copy of F all of whose edges receive a different color. This maximum, denoted by ex^∗ (n, F), is the rainbow Turán number of F, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte [Combinatorics, Probability and Computing 16 (2007)]. We determine ex^∗ (n, F) exactly when F is a forest of stars, and …

The Partition Lattice In Many Guises, Dustin G. Hedmark

Theses and Dissertations--Mathematics

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m …

Edge-Transitive Bipartite Direct Products, Cameron M. Crenshaw

Theses and Dissertations

In their recent paper ``Edge-transitive products," Hammack, Imrich, and Klavzar showed that the direct product of connected, non-bipartite graphs is edge-transitive if and only if both factors are edge-transitive, and at least one is arc-transitive. However, little is known when the product is bipartite. This thesis extends this result (in part) for the case of bipartite graphs using a new technique called "stacking." For R-thin, connected, bipartite graphs A and B, we show that A x B is arc-transitive if and only if A and B are both arc-transitive. Further, we show A x B is edge-transitive only …

Random Tropical Curves, Magda L. Hlavacek

HMC Senior Theses

In the setting of tropical mathematics, geometric objects are rich with inherent combinatorial structure. For example, each polynomial $p(x,y)$ in the tropical setting corresponds to a tropical curve; these tropical curves correspond to unbounded graphs embedded in $\R^2$. Each of these graphs is dual to a particular subdivision of its Newton polytope; we classify tropical curves by combinatorial type based on these corresponding subdivisions. In this thesis, we aim to gain an understanding of the likeliness of the combinatorial type of a randomly chosen tropical curve by using methods from polytope geometry. We focus on tropical curves corresponding to quadratics, …

---

Go to article

Sudoku Variants On The Torus, Kira A. Wyld

HMC Senior Theses

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

Classifying The Jacobian Groups Of Adinkras, Aaron R. Bagheri

HMC Senior Theses

Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through …

Combinatorial Polynomial Hirsch Conjecture, Sam Miller

HMC Senior Theses

The Hirsch Conjecture states that for a d-dimensional polytope with n facets, the diameter of the graph of the polytope is at most n-d. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in n and d on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the …

Distance Magic-Type And Distance Antimagic-Type Labelings Of Graphs, Bryan Freyberg

Dissertations, Master's Theses and Master's Reports

Generally speaking, a distance magic-type labeling of a graph G of order n is a bijection f from the vertex set of the graph to the first n natural numbers or to the elements of a group of order n, with the property that the weight of each vertex is the same. The weight of a vertex x is defined as the sum (or appropriate group operation) of all the labels of vertices adjacent to x. If instead we require that all weights differ, then we refer to the labeling as a distance antimagic-type labeling. This idea can be generalized …

Gallai-Ramsey Numbers For C7 With Multiple Colors, Dylan Bruce

Honors Undergraduate Theses

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H₁, ..., H_k, we write G → (H₁, ..., H_k), or G → (H)_k when H₁ = ··· = H_k = H, if every k-edge-coloring of G contains a monochromatic H_i in color i for some i ∈ …

On The Graceful Cartesian Product Of Alpha-Trees, Christian Barrientos, Sarah Minion

Theory and Applications of Graphs

A graceful labeling of a graph G of size n is an injective assignment of integers from the set {0,1,…,n} to the vertices of G such that when each edge has assigned a weight, given by the absolute value of the difference of the labels of its end vertices, all the weights are distinct. A graceful labeling is called an α-labeling when the graph G is bipartite, with stable sets A and B, and the labels assigned to the vertices in A are smaller than the labels assigned to the vertices in B. In this …

Edge Colorings Of K(M,N) With M+N-1 Colors Which Forbid Rainbow Cycles, Peter Johnson, Claire Zhang

Theory and Applications of Graphs

For positive integers m, n, the greatest number of colors that can appear in an edge coloring of K(m,n) which avoids rainbow cycles is m + n - 1. Here these colorings are constructively characterized. It turns out that these colorings can be encoded by certain vertex labelings of full binary trees with m + n leafs.

Distribution Of Permutation Statistics Across Pattern Avoidance Classes, And The Search For A Denert-Associated Condition Equivalent To Pattern Avoidance, Joshua Thomas Agustin Davies

Dissertations, Master's Theses and Master's Reports

We begin with a discussion of the symmetricity of $\maj$ over $\des$ in pattern avoidance classes, and its relationship to $\maj$-Wilf equivalence. From this, we explore the distribution of permutation statistics across pattern avoidance for patterns of length 3 and 4.

We then begin discussion of Han's bijection, a bijection on permutations which sends the major index to Denert's statistic and the descent number to the (strong) excedance number. We show the existence of several infinite families of fixed points for Han's bijection.

Finally, we discuss the image of pattern avoidance classes under Han's bijection, for the purpose of finding …