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Braxton Carrigan Southern Connecticut State University |
Aaron Clark Southern Connecticut State University James Hammer Cedar Crest College |
Graph Decompositions of Generalized Johnson Graphs A family of graphs, called Generalized Johnson graphs, are an abstraction of both Kneser and Johnson graphs. Given the symmetric nature of Generalized Johnson graphs, we provide various decompositions of these graphs and show non-trivial instances of the inability to decompose such graphs into triples. |
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Bradley Fain University of Delaware |
Semi-Symmetric Designs and Semi-planar Functions A function, $f$, between finite abelian groups $G$ and $H$ is said to be a planar function if for all nonidentity $a\in G$, $\Delta_{f,a}(x) = f(x+a)-f(x)$ is one-to-one. One can define an incidence structure with points set $G\times H$ by using $f$. When $f$ is planar this structure is an affine plane. In this talk I will focus on semi-planar functions of index $\lambda$, a generalization of planar functions where $\Delta_{f,a}$ is $\lambda$-to-one for all $a$. For these functions, the incidence structure is either a semi-symmetric design or a collection of isomorphic semi-symmetric designs. The main result will be a characterization of the functions $f$ for which this incidence structure is connected. |
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Carrie Finch-Smith Washington and Lee University |
Constructing Riesel and Sierpiński numbers using coverings Do you ever contemplate coverings of the integers? Have you found yourself pondering periodicity of integer sequences? Do you revel in recurrence relations? This talk about Sierpiński and Reisel numbers - what we know, what we don’t know, and what we’d like to know - has all of that and more, plus all the satisfying number theory your heart desires. |
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Brittany Gelb Muhlenberg College |
Kathryn Beck Dickinson College Lisa Cenek Amherst College Megan Cream Lehigh University |
Chorded Pancyclic Properties in Claw-Free Graphs A graph $G$ is (doubly) chorded pancyclic if $G$ contains a (doubly) chorded cycle of every possible length $m$ for $4 \leq m \leq |V(G)|$. In 2018, Cream, Gould, and Larsen completely characterized the pairs of forbidden subgraphs that guarantee chorded pancyclicity in 2-connected graphs. In our work, we show that the same pairs also imply doubly chorded pancyclicity. We further characterize conditions for the stronger property of doubly chorded $(k, m)$-pancyclicity where every set of $k$ vertices in $G$ is contained in a doubly chorded $m$-cycle for all $4 \leq m \leq |V(G)|$. In particular, we examine forbidden pairs and degree sum conditions that guarantee this recently defined cycle property. This work was completed as a part of the 2019 Lafayette College REU and supported by the NSF under grant number 1560222. |
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James Hammer Cedar Crest College |
Braxton Carrigan Southern Connecticut State University John Lorch Ball State University |
A Regional Kronecker Product and Multiple-Pair Latin Squres We develop and apply an analog of the Kronecker product, called a regional Kronecker product, that allows new, larger multiple-pair latin squares to be created from existing multiple-pair latin squares, and likewise for mutually orthogonal families of multiple-pair latin squares. We are particularly interested in applications of this product to sudoku-pair latin squares, which are a special class of multiple-pair latin squares. We show that the regional Kronecker product, when combined with new results about orthogonal sudoku-pair latin squares, has important implications for the existence and orthogonality of sudoku-pair latin squares. |
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Dean G. Hoffman Auburn University |
Sudoku gone wild! We show that a natural generalization of Sudoku puzzles is equivalent to a classic Design Theory problem. |
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Garth Isaak Lehigh University |
Catalan Multijections The Catalan numbers are ubiquitous. Richard Stanley's book on Catalan numbers lists 214 combinatorial interpretations and suggests 45,582 exercises to find bijections between these. One class of proofs that these numbers are $\frac{1}{n+1}{2n \choose n}$ focuses on the $n+1$ in the denominator. To abuse terminology, these multijective proofs are $(n+1)$ to $1$ mappings between objects counted by ${2n \choose n}$ and a Catalan family. Classic examples of this sort of proof are the Chung-Feller Theorem and the Cycle Lemma of Dvoretzky and Motzkin. Our aim is to consider other, similar proofs. We then ask the question if the other proofs are new and ‘different’ or merely ‘translations’ of the classic proofs. The framework for the answer suggests a large new class of exercises involving Catalan numbers. |
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Brian Kronenthal Kutztown University |
Alex Kodess Farmingdale State College Diego Manzano-Ruiz Kutztown University Ethan Noe Kutztown University |
The girth of real three-dimensional algebraically defined monomial graphs Consider the bipartite graph whose partite sets $P$ and $L$ are copies of $\mathbb{R}^3$ such that $(p_1,p_2,p_3)\in P$ and $[\ell_1,\ell_2,\ell_3]\in L$ are adjacent if and only if $p_2+\ell_2=p_1\ell_1$ and $p_3+\ell_3=p_1\ell_1^2$. While this graph has girth eight, replacing $p_1\ell_1$ and $p_1\ell_1^2$ with other bivariate monomials can produce graphs of different girths. In this talk, we will completely classify all such graphs by girth, as well as compare and contrast these results to what happens when $\mathbb{R}$ is replaced by other fields. |
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John Lorch Ball State University |
Generalizing Franklin's Magic Squares In the mid 1730's Benjamin Franklin was a clerk in the Pennsylvania Assembly. As a boredom-reducing form of amusement during long sessions of the Assembly, Franklin constructed interesting semi-magic squares. One such square, along with a cryptic indication of its magic properties, is shown below. $$ \begin{array}{|c | c | c | c | c | c | c | c|} \hline 52 & 61 & 4 & 13 & 20 & 29 & 36 & 45 \\ \hline 14 & 3 & 62 & 51 & 46 & 35 & 30 & 19 \\ \hline 53 & 60 & 5 & 12 & 21 & 28 & 37 & 44 \\ \hline 11 & 6 & 59 & 54& 43 & 38 & 27 & 22 \\ \hline 55 & 58 & 7 & 10 & 23 & 26 & 39& 42 \\ \hline 9 & 8& 57 & 56 & 41& 40 & 25 & 24 \\ \hline 50 & 63 & 2 & 15 & 18 & 31 & 34 & 47 \\ \hline 16 & 1 & 64 & 49 & 48 & 33 & 32& 17 \\ \hline \end{array} \qquad \begin{array}{|c | c | c | c | c | c | c | c|} \hline \ & \ & \ & 13 & 20 & \ & \ & \ \\ \hline 14 & \ & \ & 51 & 46 & \ & \ & 19 \\ \hline \ & 60 & \ & \ & \ & \ & 37 & \ \\ \hline \ &\ & 59 & \ & \ & 38 & \ & \ \\ \hline \ & \ & \ & 10 & 23 & \ & 39& \ \\ \hline \ & \ & \ & \ & \ & \ & 25 & \ \\ \hline 50 & 63 & 2 & 15 & \ & \ & 34 & \ \\ \hline \ & \ & \ & \ & \ & \ & 32 & \ \\ \hline \end{array} $$ Franklin is known to have produced two such squares of order $8$ (i.e., $8\times 8$ arrays) and one square of order $16$ with similar properties. In fact, semi-magic squares possessing the special properties exhibited in Franklin's squares exist in order $8k$, where $k$ is any positive integer. In this presentation we review Franklin's magic squares, we identify magic properties that should be possessed by a Franklin square of order $kp^3$ where $p$ is an odd prime, and we construct such squares. |
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Michael Mossinghoff Center for Communications Research |
Timothy Trudgian UNSW Canberra at ADFA |
A Tale of Two Omegas On average, would you expect there to be more positive integers $n\leq x$ with an even number of prime divisors, counting multiplicities, or with an odd number, as $x$ grows large? What if instead one counted the number of distinct prime divisors of each integer? Would this change your answer? We describe some investigations of these questions, and how they are related to a number of open problems in number theory, including the Riemann hypothesis. |
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Viorel Niţică West Chester University |
Torok Andrei University of Houston |
About some relatives of Palindromes In this talk we introduce two new classes of integers. The first class consists of all numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of digits of $N$, added to the reversal of the sum, gives $N$. The second class consists of all numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of the digits of $N$, multiplied by the reversal of the sum, gives $N$. All palindromes that either have an even number of digits or an odd number of digits and the middle digit even belong to the first class, and all squares of palindromes with at least two digits belong to the second class. These classes contain and are strictly larger than the classes of $b$-ARH numbers, respectively $b$-MRH numbers introduced in a paper by Niţică. We show many examples of such numbers and ask several questions that may lead to future research. In particular, we try to clarify some of the relationships between these classes of numbers and the well studied class of $b$-Niven numbers. Most of our results are true in a general numeration base. |
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Caitlin Owens DeSales University |
Rey Anaya Moravian College James Hammer Cedar Crest College Madeline Kohutka Cedar Crest College Emily Rombinson Mount Holyoke College Chen Sun Muhlenberg College |
Color Me Proper A proper edge coloring of a graph is an edge coloring such that no two incident edges in the graph share the same color. The proper connection of a graph is a variation of proper edge coloring introduced by Borozan, et al. According to their definition, an edge colored graph is said to be properly connected if there exists a properly colored path between every pair of vertices. They define the proper connection number of a graph, denoted $pc(G)$, as the minimum number of colors, $k$, such that there exists a $k$ edge coloring of $G$ which is properly connected. Motivated by the work on proper connection, my Summer 2019 REU group and I extended this idea by requiring that every shortest path between each pair of vertices be properly colored. We call such a coloring a very strong shortest path coloring. Furthermore, we define the very strong proper connection number of a graph, denoted $vspc(G)$, as the minimum number of colors required to edge color a graph $G$ such that the coloring is a very strong shortest path coloring. In this talk, I will discuss our results on the very strong proper connection number for certain classes of graphs. |
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Andrew Owens Widener University |
Paul Horn University of Denver Dean Hoffman Auburn University Peter Johnson Auburn University |
Edge Colorings Forbidding Rainbow Cycles It is well known that the greatest number of colors appearing in a rainbow-cycle-forbidding edge coloring of a connected graph on $n$ vertices is $n-1$. Such an edge coloring is known as a JL-coloring. In previous work it has been shown that for graphs in certain classes, these colorings are all obtainable in a certain way that permits classification: for instance, it is known that the essentially different colorings of $K_n$ with $n > 1$ are in one-to-one correspondence with isomorphism classes of full binary trees with $n$ leafs. We have defined a Standard Construction for JL-colorings (which we derived from the previous results) and we have shown that any JL-coloring of a connected graph is produced by this Standard Construction. Furthermore, every JL-coloring has a monochromatic edge cut. We also state some results on the sharpness of this result: specifically, what can we say about the number of colors used in an edge coloring that forbids rainbow cycles and monochromatic cuts. |
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Wing Hong Tony Wong Kutztown University of Pennsylvania |
Characterization of super totient numbers and applications in graph labeling Let $n$ be a positive integer, and let $R(n)$ be the set of positive integers less than $n$ and relatively prime to $n$. If there exists a partition of $R(n)$ into subsets $A$ and $B$ such that the sum of the elements in $A$ is equal to that in $B$, then $n$ is called a super totient number. In this talk, we completely characterize all super totient numbers. Furthermore, we define and investigate on two new concepts related to graph labeling: a restricted super totient labeling and the super totient index of graphs. In particular, we completely characterize all trees that admit a restricted super totient labeling, and provide the super totient index of several families of graphs. |