The distinguishing chromatic number $\chi_{_D}(G)$ of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number $\chi(\cdot)$. Our main result is that for every graph G, there exists a constant $d_G$ such that for all $d\geq d_G$ the distinguishing chromatic number of $G^d$ is at most $\chi(G) +1$, where $G^d$ is the Cartesian product of d copies of G. We also prove that for $d\geq5$, the Cartesian product of d complete graphs has distinguishing chromatic number at most one more than the corresponding chromatic number, and we determine the distinguishing chromatic number of hypercubes exactly.

Given a partition of the vertices of a graph into two sets, a flip is a move of a vertex from its own set to the other, under the condition that it has more incident edges to vertices in its own set than in the other. Every sequence of flips eventually produces a bipartite subgraph capturing more than half of the edges in the graph. Each flip gains at least one edge. For an n-vertex loopless multigraph, we show that there is always a sequence of at most $n/2$ flips that cannot be extended, and we construct a graph having a sequence of $\frac2{25}(n^2+n-31)$ flips.