Abstracts for Invited Speakers

s for Invited Speakers What follows is a list of abstracts for the invited speakers in the order that they will be presented. These can also be found individually via the conference website’s participant list. Extending Fisher’s inequality to coverings Daniel Horsley School of Mathematical Sciences, Monash University, Melbourne, Australia [email protected] A (v, k, λ)-design is a collection of k-element subsets, called blocks, of a v-set of points such that each pair of points occurs in exactly λ blocks. Fisher’s inequality is a classical result that states that every nontrivial (v, k, λ)-design has at least v blocks (equivalently, has v ≥ k(k−1) λ +1). An elegant proof of Fisher’s inequality, due to Bose, centres on the observation that if X is the incidence matrix of a nontrivial design, then XX is nonsingular. This talk is about extending this proof method to obtain results on coverings. A (v, k, λ)-covering is a collection of k-element blocks of a v-set of points such that each pair of points occurs in at least λ blocks. Bose’s proof method can be extended to improve the classical bounds on the number of blocks in a (v, k, λ)-covering when v < k(k−1) λ + 1. Specifically, this is accomplished via bounding the rank of XX , where X is the incidence matrix of a (v, k, λ)-covering, using arguments involving diagonally dominant matrices and m-independent sets in multigraphs. A Tale of Universal Cycles in Two REU-Environments Anant Godbole Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614 [email protected] I regularly supervise undergraduate research projects as part of my tenured position at ETSU, where we require undergraduate research of all our majors via the course MATH 4010. At the same time I have run an NSF-sponsored REU site during most years since 1991. This talk will outline the symbiotic nature of these undertakings. I will provide concrete examples of how great NSF-REU work led to important work by an ETSU undergraduate and, conversely, how a project started in MATH 4010 led to important REU work in the summer. In both cases, the students in question were working on Universal Cycles, introduced into the literature by Chung, Diaconis, and Graham. Edge colouring multigraphs Penny Haxell Combinatorics and Optimization Department, University of Waterloo, Waterloo, Ontario, Canada [email protected] While edge colouring in graphs is well understood (by Vizing’s classical theorem), the chromatic index χ′(G) of a multigraph G can fall anywhere in the range [∆(G), 3∆(G)/2] (where ∆(G) denotes the maximum degree of G), or in terms of the maximum edge multiplicity μ(G) the range [∆(G),∆(G)+μ(G)]. We discuss results showing that if χ′(G) is significantly larger than ∆(G), then G contains a small subgraph that is very dense. This idea is at the core of the famous open problem of Goldberg and (independently) Seymour, which seeks to identify the properties of G that influence χ′(G). 3-Flows with Large Support Jessica McDonald Department of Mathematics and Statistics, Auburn University, Auburn, AL 36830 [email protected] Tutte’s 3-Flow Conjecture says that every 4-edge-connected graph should have a nowhere-zero 3-flow. The 4-edge-connectivity assumption cannot be weakened—K4 is an example of a 3-edge-connected graph that does not have a nowhere-zero 3-flow. However, K4 is minimal in the sense that K − 4 has a nowhere-zero 3-flow. Since K4 has 6 edges in total, this means that we are able to give K4 a 3-flow in which 5/6 of the edges are nonzero. With DeVos, Pivotto, Rollova, and Samal, we can show that this is the worst case in general—that is, if G is any 3-edge-connected graph, then G has a 3-flow with support size at least 56 |E(G)|. As a corollary, this implies that every planar graph has an assignment of three colours to its vertices so that at most a sixth of its edges join vertices of the same colour. Hamilton decompositions Barbara Maenhaut School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland, Australia [email protected] The problem of partitioning the edge set of a graph into Hamilton cycles is a classical problem, with results dating back to the work of Walecki in the 1890s. Over the last 50 years, much work has been done on several open conjectures and problems that deal with Hamilton decompositions of certain families of graphs. I will survey some of this work and discuss a few recent developments, especially on Hamilton decompositions of vertex-transitive graphs, line graphs, and infinite graphs. Abstracts for Contributed Talkss for Contributed Talks What follows is a list of abstracts for the contributed talks organized by lead presenter’s last name (as listed in the schedule). These can also be found individually via the conference website’s participant list. Combinatorial-based Pairwise Event Sequence Generation for Automated GUI Testing of Android Apps David Adamo*, Renée Bryce Department of Computer Science and Engineering, University of North Texas, Denton, TX 76203 [email protected] Mobile apps are Event-Driven Systems (EDS) that often rely on Graphical User Interfaces (GUI) as the primary means of interaction with end-users. These systems are tested using event sequences that exercise the system’s functionality while covering as much of its source code as possible. In this work, we develop a greedy pairwise approach to automate event sequence generation for Android app testing. The proposed approach dynamically traverses an app by identifying and executing GUI events that maximize the coverage of pairwise event combinations. We implement two variants of a greedy pairwise algorithm. The first considers the order of the event pairs while the second one does not. We conduct experiments on five Android apps by comparing the proposed approach to random event sequence generation. The results show that given the same parameters, test suites generated using a greedy pairwise approach tend to achieve higher block coverage than test suites with randomly generated event sequences. The results also show that pairwise test suites tend to find a higher number of faults than randomly generated test suites. However, the results show that there was no significant difference between pairwise test suites generated using ordered pairs and pairwise test suites generated using unordered pairs in terms of block coverage and number of faults found. On Enumeration of Paths in Catalan–Schröder Lattices Max A. Alekseyev Department of Mathematics, The George Washington University, Washington, DC 20052 [email protected] We address the problem of enumerating paths in square lattices, where allowed steps include (1, 0) and (0, 1) everywhere, and (1, 1) above the diagonal y = x. We consider two such lattices differing in whether the (1, 1) steps are allowed along the diagonal itself. Our analysis leads to explicit generating functions and an efficient way to compute terms of many sequences in the Online Encyclopedia of Integer Sequences, proposed by Clark Kimberling over a decade and a half ago. The work is supported by the National Science Foundation under grant No. IIS-1462107. Avoiding subsystems in cycle systems John Asplund*, Michael Schroeder, Venkata Dinavahi Department of Technology and Mathematics, Dalton State College, Dalton, GA 30720; Department of Mathematics, Marshall University, Huntington, WV 25755; Department of Mathematics, University of Findlay, Findlay, OH 45840 [email protected] A k-cycle decomposition of G is a partition of the edge set of G such that each element of the partition induces a k-cycle. If G = Kn then it is called a k-cycle system of order n. The necessary and sufficient conditions for the existence of a k-cycle system of order n have already been determined. We aim to take this one step further. We will show how minute changes to systems can impact the structure of the k-cycle system in interesting ways. In particular, this talk will show there exists a k-cycle system P of order n such that no subset of P forms a k-cycle system of order t where 2 < t < n and both n and k are odd; if we can show this, we say that P contains no subsystems. The Symmetric Arctic Rank of Boolean Matrices and the Clique Covering Content of Graphs LeRoy B. Beasley Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900 [email protected] Let G be a simple undirected graph on n vertices and let A = A(G) be the (Boolean) (0, 1) adjacency matrix of G. Let ρsa(A) be the minimum |B| over all possible factorizations of A of the form BB for some (Boolean) n × k (0, 1) matrix B and some k where |B| denotes the number of nonzero entries in B. This function, ρsa(A), is called the symmetric arctic rank of A. Let C(G) denote the set of all clique covers of G, and let C(G) be the minimum over C(G) of sum of the orders of the cliques composing the clique cover. This function, C(G), is called the clique cover content of G. It is shown that C(G) = ρsa(A(G)) and some properties and problems are presented. On Path-Hamiltonian Tournaments Zhenming Bi Department of Mathematics, Western Michigan University, [email protected] A Hamiltonian graph G is j-path Hamiltonian if every path of order j can be extended to a Hamiltonian cycle of G. The Hamiltonian extension number he(G) of G is the largest integer k such that G is j-path Hamiltonian for every integer j with 1 ≤ j ≤ k. Hamiltonian extension in graphs has been studied. Here, we study corresponding concepts for Hamiltonian tournaments. Results and open questions are presented in this area of research. This is joint work with F. Fujie and P. Zhang. Re-Visiting Rainbow Connection in Graphs Alexis Byers Department of Mathematics, Western Michigan University [email protected] A rainbow coloring of a connected graph G is an edge coloring of G, where adjacent edges may be colored the same, with the property that for every two vertices u and v of G, there exists a u− v rainbow path (no two edges of the path are colored the same). The minimum number of colors in a rainbow coloring of G is the rainbow connection number of G. This topic has been studied by many. We revisit this concept from a different point of view. Recent results and problems in this area of research are presented. This is joint work with Z. Bi, G. Chartrand, and P. Zhang. Perfect matchings in bipartite graphs, Mersenne primes and 2-rooted primes Sunil Chebolu*, Keir Lockridge, Gaywalee Yamskulna Department of Mathematics, Illinois State University, Normal, IL 61790 [email protected] We give graph theoretic characterizations of two important classes of prime numbers: Mersenne primes and primes p for which 2 generates the multiplicative group of the field of p elements. Our characterizations of these prime numbers are based on perfect matchings in circulant bipartite graphs on 2p vertices. [arXiv:1404.4096] On Balanced Factorial Designs of Resolution Ten and Balanced Arrays of Strength Nine D. V. Chopra*, R. M. Low Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS 67260, USA [email protected] Balanced arrays (B-arrays) are generalizations of orthogonal arrays (O-arrays) and have been extensively used in the construction of balanced factorial designs. An array T with two levels (say, 0 and 1), m constraints (factors), and N treatment-combinations (runs) is merely a matrix T of size (m ×N) with two elements 0 and 1. T is said to be of strength t (≤ m) if, in each t-rowed submatrix T ∗ of T , the following condition is satisfied: every vector α of T ∗ of weight i (0 ≤ i ≤ t) (the weight of a vector α is defined to be the number of 1’s in it) appears with the same frequency μi (say, 0 ≤ i ≤ t). The set {m;μ0, μ1, . . . , μt} is called the set of parameters for the array T . If μi = μ, then T is called an O-array with parameter μ. It is quite obvious that N is known if the μi’s are given. It is well known that B-arrays of strength t, under certain conditions, give rise to balanced factorial designs of resolution (t + 1) in which each of the m factors is at two levels, denoted in design theory by 2 designs. In this paper, we derive some existence conditions for B-arrays with t = 9. We also use these conditions for obtaining max(m) for a given set (μ0, μ1, . . . , μ9). Dominating Parameters of Certain Graphs Alpeshkumar Dhorajia Birla Institute of Technology and Science, K K Birla Goa Campus, Goa-403726, India [email protected] In this talk we will discuss about various number of dominating parameters including the dominating number, independent dominating number, clique dominating number, connected dominating number, strong dominating number and weak dominating number of total graph of Zn × Zm. Cz frames of M(b, n), where z is even M. Tieyemer, D. Prier, Chandra Dinavahi* Department of Mathematics, The University of Findlay, Findlay, OH 45840 [email protected] Let M(b, n) be the complete multipartite graph with b parts B0, . . . , Bb−1 of size n. A z-cycle system of M(b, n) is said to be a Cz-frame if the z-cycles can be partitioned into sets S1, . . . , Sk such that for 1 ≤ j ≤ k, where Sj induces a 2-factor of M(b, n) − Bi for some i ∈ Zb. The existence of a Cz-frame of M(b, n) has been settled when z ∈ {3, 4, 5, 6}. Here, we consider Cz-frames when z ≥ 8 is even. Tridiagonal and Hessenberg Matrices Representing Recursive Number Sequences Ximing Dong*, Tian-Xiao He Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61701 [email protected] We present a general method for constructing sequences of certain tridiagonal matrices from a second-order recursive number sequence with initial condition a0 = 0 and a1 = 1, so that the permanents of the matrices give values of the sequence elements. Then the result is extended to the case of the second-order recursive number sequences with arbitrary initials by using a type of tridiagonal matrices. Finally, the permanents of certain upper-Hessenberg matrices are used to represent some n-order recurrence number sequences. Square-free Rank of Integers Roger B. Eggleton*, Jason S. Kimberley, James A. MacDougall University of Newcastle, Callaghan, NSW 2308, Australia [email protected] For any positive integer n, there are positive integers a, b such that a is the largest square divisor of n, and n = ab: the square-free rank of n is the number of prime divisors of b. For instance, 242 = 11 · 2 and 243 = 9 · 3 lie in a run of 5 consecutive integers all of square-free rank 1. Below 10, the longest run of integers with constant square-free rank has 20 members, all of square-free rank 3. For each k ≥ 1, we give an explicit upper bound on the size of any run of integers of square-free rank k, and show (at least for k ≤ 7) there are infinitely many pairs of consecutive integers of square-free rank k. For distinct primes p, q there is a smallest positive integer d(p, q) such that 0 < ap− bq = d(p, q) ≤ p holds for infinitely many integers a, b. We show the sequence ( 3n | n ∈ N ) has no member in the interval [ 2a, 2(a+ 1) ] with a ∈ N if and only if a is in the Beatty sequence ( ⌊( 3 + √ 6 ) n ⌋ | n ∈ N ) . Finally, we study sequences (a1, a2, . . . , ak) for square-free rank 1 integers 2a1 > 3a 2 2 > · · · > pkak with pi(ai + 1) > 2a1 for 1 < i ≤ k, and the corresponding sequences in which all the inequalities are reversed and pi(ai − 1) < 2a1 for 1 < i ≤ k. Decompositions of Complete Digraphs into Small Tripartite Digraphs Steven DeShong∗, Alexander Fischer*, Lawrence Teng* Virginia Polytechnic Institute and State University, Blacksburg, VA 24061; Jacobs High School, Algonquin, IL 60102; University of Michigan, Ann Arbor, MI 48109, USA [email protected] The paw graph consists of a triangle with a pendant edge attached to one of the three vertices. We obtain a multigraph by adding exactly one repeated edge to the paw. Now, let D be a directed graph obtained by orientating the edges of that multigraph. For 12 of the 18 possibilities for D, we establish necessary and sufficient conditions on n for the existence of a (K∗ n, D)-design. Partial results are given for the remaining 6 possibilities for D. This research is supported by grant number A1359300 from the Division of Mathematical Sciences at the National Science Foundation. Incomplete tournaments with handicap two Dalibor Froncek Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN [email protected] A d-handicap distance antimagic labeling of a graph G with vertex set V = {x1, x2, . . . , xn} and edge set E is a bijection f : V → {1, 2, . . . , n} with induced function w : V → N defined as