4−Equitable Tree Labelings

We assign the labels {0,1,2,3} to the vertices of a graph; each edge is assigned the absolute difference of the incident vertices’ labels. For the labeling to be 4−equitable, we require the edge labels and vertex labels to each be distributed as uniformly as possible. We study 4−equitable labelings of different trees and prove all cater-pillars, symmetric generalized n−stars (or symmetric spiders), and complete n −ary trees for all n ∈ N are 4−equitable

The Game of Cycles for Grids and Select Theta Graphs

We are investigating who has the winning strategy in a game in which two players take turns drawing arrows trying to complete cycle cells in a graph. A cycle cell is a cycle with no chords. We examine game boards where the winning strategy was previously unknown. Starting with a $C_{5}$ sharing two consecutive edges with a $C_{7}$ we solve multiple classes of graphs involving "stacked" polygons. We then expand upon and improve previous theorems and conjectures, and offer some new directions of research related to the Game of Cycles. The original game was described by Francis Su in his book Mathematics for Human Flourishing. The first results on the game were published in The Game of Cycles arXiv:arch-ive/04.00776.Comment: 16 pages, 17 figure

4−Equitable Tree Labelings

We assign the labels {0,1,2,3} to the vertices of a graph; each edge is assigned the absolute difference of the incident vertices’ labels. For the labeling to be 4−equitable, we require the edge labels and vertex labels to each be distributed as uniformly as possible. We study 4−equitable labelings of different trees and prove all cater-pillars, symmetric generalized n−stars (or symmetric spiders), and complete n −ary trees for all n ∈ N are 4−equitable

The Cover Pebbling Number of Graphs

A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of the graph we can eventually put a pebble on every vertex simultaneously. We find the cover pebbling numbers of trees and some other graphs. We also consider the more general problem where (possibly different) given numbers of pebbles are required for the vertices.Comment: 12 pages. Submitted to Discrete Mathematic

On the Bollob\\u27as inequality

In this thesis we are concerned with the poset ${\cal P}(n)={\cal P}(\{1,2,\...,n\} ),$ the power set of $\lbrack n\rbrack=(\{1,2,\..., n\},$ ordered by inclusion. Consider a collection of pairs of sets, \{(A\sb{i},B\sb{i})\}\sbsp{i=1}{m} in ${\cal P}(n)$ with the property that \Vert A\sb{i}\Vert=a and \Vert B\sb{i}\Vert=b, for each $i=1,\..., m.$ The Bollobas inequality gives a bound on m in the case when A\sb{i}\cap B\sb{j} is empty if and only if i = j. We present the history of the problem and its generalizations, and we make new contributions in two different directions. First we extend a result of Furedi and answer a question of Babai by giving an asymptotically sharp bound on m in the case when \vert A\sb{i}\cap B\sb{i}\vert\le p for all i, and \vert A\sb{i}\cap B\sb{j}\vert\ge q for all $i\ne j.$ Then we consider two set-systems ${\cal A}$ and ${\cal B}$ in the powerset ${\cal P}(n)$ with the property that for each $A\in{\cal A}$ there exists a unique $B\in{\cal B}$ such that $A\subset B.$ Ahlswede and Cai proved an inequality about such systems which is a generalization of the Bollobas inequality. We characterize the structure of the extremal cases, and exhibit a one-to-one correspondence between the extremal cases and certain matroids.

Challenges in Promoting Undergraduate Research in the Mathematical Sciences

We describe the challenges in promoting undergraduate research in the mathematical sciences. The challenges are grouped in regards to the population that research is promoted to: students, faculty and administrators. For each category, we provide some suggestions for overcoming the challenges taking into account the variety of institutions involved