In this talk, we present a survey of the currently best known upper bounds on the domination number $\gamma(G)$ and total domination number $\gamma_t(G)$ of a graph $G$ in terms of its order $n$ and minimum degree $\delta$. We focus mainly on results for small values of $\delta$, namely $\delta \le 6$. We present a sketch proof, that combines an algorithmic approach using vertex weighting arguments together with discharging methods, of the result that if $\delta = 6$, then $\gamma(G) \le \frac{127}{418}\, n \approx 0.30382775 \, n$. We also present a sketch proof, that uses transversals in hypergraphs, of the result that if $\delta = 6$, then $\gamma_t(G) \le \frac{4}{11}\, n \approx 0.363636 \, n$.

One of the central objects of interest in additive combinatorics is the sumset $A+B= \{a+b:a \in A, b \in B\}$ of two sets $A,B$ of integers.

Our main theorem, which improves results of Green and Morris, and of Mazur, implies that the following holds for every fixed $\lambda > 2$ and every $k>(\log n)^4$: if $\omega$ goes to infinity as $n$ goes to infinity (arbitrarily slowly), then almost all sets $A$ of $[n]$ with $|A| = k$ and $|A + A| < \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega$.

This is joint work with Marcelo Campos, Mauricio Collares, Rob Morris and Victor Souza.

We study two eternal, or dynamic, variants of graph coloring. In the first version, starting with a proper coloring of graph $G$, an infinite sequence of requests occurs, each of which forces a vertex to change colors while maintaining a proper coloring. The minimum number of colors needed for $G$ over any such sequence is the eternal chromatic number of $G$. Bounds on the parameter are given.

The second problem we consider is a version of game coloring. Recall that game coloring is a two-player game in which each player properly colors one vertex of a graph at a time until all the vertices are colored. In the eternal version of game coloring, the vertices are colored and then subsequently re-colored from a color set over an infinite sequence of rounds. In a given round, each vertex is colored, or re-colored, once, so that a proper coloring is maintained. Player 1 wants to maintain a proper coloring forever, while Player 2 wants to force the coloring process to fail. The eternal game chromatic number of a graph $G$ is the minimum number of colors needed in the color set so that Player 1 can always win the game on $G$. Bounds are shown on some elementary classes of graphs. A number of conjectures and questions are posed for each version of the problem.