As the underlying configuration behind many elegant finite structures, partial difference sets have been intensively studied in design theory, finite geometry, coding theory, and graph theory. Over the past three decades, there have been numerous constructions of partial difference sets in abelian groups with high exponent, accompanied by numerous very different and delicate techniques. Surprisingly, we manage to unify and extend a great many previous constructions in a common framework, using only elementary methods. The key insight is that, instead of focusing on one single partial difference set, we consider a packing of partial difference sets, namely, a collection of disjoint partial difference sets in a finite abelian group. This conceptual shift leads to a recursive lifting construction of packings in abelian groups with increasing exponent.

This is joint work with Jonathan Jedwab.

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $\gamma(G)$, is the minimum size of a dominating set of $G$. The independent domination number of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent. Let $G$ be a connected $k$-regular graph that is not $K_{k, k}$ where $k\geq 4$. We prove that $i(G)\le \frac{k-1}{2k-1}|V(G)|$, which is tight for $k = 4$. This answers a question by Goddard et al.(2012) in the affirmative. We also present a survey and a recent result on the independent domination number of a cubic graph.

We present recent results on generalized Mahonian statistics and factorizations of the full cycle. We begin with a review of classical Mahonian statistics, specifically the major index and inversion statistic, on permutations, and then discuss how these statistics apply to more general objects such as trees and cacti. Our results show how these statistics are connected with natural statistics on minimal factorizations of the (canonical) full cycle into cycles of fixed length. We establish this connection through a direct bijection that involves elementary operations on planar objects. Various properties of these statistics will also be presented (for example, symmetries of these statistics and a new hook length formula related to them).

This is joint work with J. Irving.