Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Barry Balof

• TITLE: Baire One Functions
AUTHOR: Johnny Hu
ABSTRACT: This paper gives a general overview of Baire one functions, including examples as well as several interesting properties involving bounds, uniform convergence, continuity, and $F_\sigma$ sets. We conclude with a result on a characterization of Baire one functions in terms of the notion of first return recoverability, which is a topic of current research in analysis.

• TITLE: Upper bounds on the $L(2;1)$-labeling Number of Graphs with Maximum Degree $\Delta$
AUTHOR: Andrew Lum
ABSTRACT: $L(2;1)$-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs to model the channel assignment problem proposed by Fred Roberts [Ro, 1988]. An $L(2;1)$-labeling of a simple graph $G$ is a nonnegative integer-valued function $f : V(G)\rightarrow \{0,1,2,\ldots\}$ such that, whenever $x$ and $y$ are two adjacent vertices in $V(G)$, then $|f(x)-f(y)|\geq 2$, and, whenever the distance between $x$ and $y$ is 2, then $|f(x)-f(y)|\geq 1$. The $L(2;1)$-labeling number of $G$, denoted $\lambda(G)$, is the smallest number $m$ such that $G$ has an $L(2;1)$-labeling with no label greater than $m$. Much work has been done to bound $\lambda(G)$ with respect to the maximum degree $\Delta$ of $G$ ([Cha, 1996], [Go, 2004], [Gr, 1992], [Kr, 2003], [Jo, 1993]). Griggs and Yeh [Gr, 1992] conjectured that $\lambda \leq \Delta^2$ when $\Delta \geq 2$.

In §1, we review the basics of graph theory. This section is intended for those with little or no background in graph theory and may be skipped as needed. In §2, we introduce the notion of $L(2;1)$-labeling. In §3, we give the labeling numbers for special classes of graphs. In §4, we use the greedy labeling algorithm to establish an upper bound for $\lambda$ in terms of $\Delta$. In §5, we use the Chang-Kuo algorithm to improve our bound. In §6, we prove the best known bound for general graphs.

• TITLE: Bijections on Riordan Objects(voted outstanding senior project)
AUTHOR: Jacob Menashe
ABSTRACT: The Riordan Numbers are an integer sequence closely related to the well-known Catalan Numbers . They count many mathematical objects and concepts. Among these objects are the Riordan Paths, Catalan Partitions, Interesting Semiorders, Specialized Dyck Paths, and Riordan Trees. That these objects have been shown combinatorially to be counted by the same sequence implies that a bijection exists between each pair. In this paper we introduce algorithmic bijections between each object and the Riordan Paths. Through function composition, we thus construct 10 explicit bijections: one for each pair of objects.

• TITLE: The Problem of Redistricting: the Use of Centroidal Voronoi Diagrams to Build Unbiased Congressional Districts
AUTHOR: Stacy Miller
ABSTRACT: This paper is a development of the use of MacQueen’s method to draw centroidal Voronoi diagrams as apart of the redistricting process. We will use Washington State as an example of this method. Since centroidal Voronoi diagrams are inherently compact and can be created by an unbiased process, they could create congressional districts that are not only free from political gerrymandering but also appear to the general public as such.