The two-coloring number of graphs, which was originally introduced in the study of the game chromatic number, also gives an upper bound on the degenerate chromatic number as introduced by Borodin. It is proved that the two-coloring number of any planar graph is at most nine. As a consequence, the degenerate list chromatic number of any planar graph is at most nine. It is also shown that the degenerate diagonal chromatic number is at most 11 and the degenerate diagonal list chromatic number is at most 12 for all planar graphs.

The power system monitoring problem asks for as few as possible measurement devices to be put in an electric power system. The problem has a graph theory model involving power dominating sets in graphs. The power domination number $\gamma_P(G)$ of G is the minimum cardinality of a power dominating set. Dorfling and Henning [Discrete Appl. Math., 154 (2006), pp. 1023–1027] determined the power domination number of the Cartesian product of paths. In this paper the power domination number is determined for all direct products of paths except for the odd component of the direct product of two odd paths. For instance, if n is even and C a connected component of $P_m\times P_n$, where m is odd or $m\geq n$, then $\gamma_P(C)=\left\lceil n/4 \right\rceil$. For the strong product we prove that $\gamma_P(P_n \boxtimes P_m) = \max\{\lceil n/3\rceil, \lceil (n+m-2)/4\rceil\}$, unless $3m-n-6 \equiv 4\pmod 8$. The power domination number is also determined for an arbitrary lexicographic product.

We define the neighborhood of an r-ball $B(u,r)$ centered on $u\in G$ as the set of all vertices x, such that $d(u,x)=r+1$, and denote it by $N(u,r)$. We call a set X of pairwise disjoint r-balls an optimal local structure for $B(u,r)$ if $N(u,r)\subset \bigcup X$ and no r-ball from X intersects $B(u,r)$. We prove the nonexistence of an optimal local structure in $G=C_{q_1} \square C_{q_2} \square \cdots \square C_{q_n}$ for any $B(u,r)\subset G$, where $n\geq 3,$ $r\geq n$, and $q_i\geq 2r+1$ for $i=1,\ldots,n$. In particular, this confirms the nonexistence of perfect Lee codes with parameters $n\geq 3,$ $e \geq n$, and $q\geq 2e+1$. We also prove that if $q_i$ is even for $i=1,\ldots,n$, $\sum_{i=1}^n q_i/2$ is odd, and $2r+1=\sum_{i=1}^n q_i/2$, then for every r-ball in G there is an optimal local structure.