abstract: This thesis focuses on sequencing questions in a way that provides students with manageable steps to understand some of the fundamental concepts in discrete mathematics. The questions are aimed at younger students (middle and high school aged) with the goal of helping young students, who have likely never seen discrete mathematics, to learn through guided discovery. Chapter 2 is the bulk of this thesis as it provides questions, hints, solutions, as well as a brief discussion of each question. In the discussions following the questions, I have attempted to illustrate some relationships between the current question and previous questions, explain the learning goals of that question, as well as point out possible flaws in students' thinking or point out ways to explore this topic further. Chapter 3 provides additional questions with hints and solutions, but no discussion. Many of the questions in Chapter 3 contain ideas similar to questions in Chapter 2, but also illustrate how versatile discrete mathematics topics are. Chapter 4 focuses on possible future directions. The overall framework for the questions is that a student is hosting a birthday party, and all of the questions are ones that might actually come up in party planning. The purpose of putting it in this setting is to make the questions seem more coherent and less arbitrary or forced. / Dissertation/Thesis / Masters Thesis Mathematics 2014

A design is a set of points, V, together with a set of subsets of V called blocks. A classic type of design is a balanced incomplete block design, where every pair of points occurs together in a block the same number of times. This ‘balanced’ condition can be replaced with other properties. An adesign is a design where instead every pair of points occurs a different number of times together in a block. The number of times a specified pair of points occurs together is called the pair frequency. Here, a special type of adesign is explored, called a Sarvate-Beam design, named after its founders D.G. Sarvate and W. Beam. In such an adesign, the pair frequencies cover an interval of consecutive integers. Specifically the existence of Sarvate-Beam group divisible designs are investigated. A group divisible design, in the usual sense, is a set of points and blocks where the points are partitioned into subsets called groups. Any pair of points contained in a group have pair frequency zero and pairs of points from different groups have pair frequency one. A Sarvate-Beam group divisible design, or SBGDD, is a group divisible design where instead the frequencies of pairs from different groups form a set of distinct nonnegative consecutive integers. The SBGDD is said to be uniform when the groups are of equal size. The main result of this dissertation is to completely settle the existence question for uniform SBGDDs with blocks of size three where the smallest pair frequency, called the starting frequency, is zero. Higher starting frequencies are also considered and settled for all positive integers except when the SBGDD is partitioned into eight groups where a few possible exceptions remain. A relationship between these designs and graph decompositions is developed and leads to some generalizations. The use of matrices and linear programming is also explored and give rise to related results. / Graduate

Suppose there is a transmitter located at each vertex of a graph G. A k-limited broadcast on G is an assignment of the integers 0,1,...,k to the vertices of G. The integer assigned to the vertex x represents the strength of the broadcast from x, where strength 0 means the transmitter at x is not broadcasting. A broadcast of positive strength s from x is heard by all vertices at distance at most s from x. A k-limited broadcast is called dominating if every vertex assigned 0 is within distance d of a vertex whose transmitter is broadcasting with strength at least d. The k-limited broadcast domination number of G is the minimum possible value of the sum of the strengths of the broadcasts in a k-limited dominating broadcast of G. We establish upper and lower bounds for the 2-limited broadcast domination number of various grid graphs, in particular the Cartesian products of two paths, a path and a cycle, and two cycles. The upper bounds are derived by explicit broadcast constructions for these graphs. The lower bounds are obtained via linear programming duality by finding lower bounds for the fractional 2-limited multipacking number of these graphs. Finally, we present an algorithm to improve the lower bound for the 2-limited broadcast domination number of special sub-families of grids. We conclude this thesis with suggested open problems in broadcast domination and multipackings. / Graduate