Domination game (Brešar et al. in SIAM J Discrete Math 24:979-991, 2010) and total domination game (Henning et al. in Graphs Comb 31:1453-1462 (2015) are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than 6/7 of the order of a graph.
COBISS.SI-ID: 18819673
If $S=(a_1,a_2,\ldots)$ is a non-decreasing sequence of positive integers, then an $S$-packing coloring of a graph $G$ is a partition of $V (G)$ into sets $X_1,X_2, \ldots$ such that for each pair of distinct vertices in the set $X_i$, the distance between them is larger than $a_i$. If there exists an integer $k$ such that $V(G)=X_1\cup \cdots \cup X_k$, then the partition is called an $S$-packing $k$-coloring. The $S$-packing chromatic number of $G$ is the smallest $k$ such that $G$ admits an $S$-packing $k$-coloring. If $a_i=i$ for every $i$, then the terminology reduces to packing colorings and packing chromatic number. Since the introduction of these generalizations of the chromatic number in 2008 more than fifty papers followed. Here we survey the state of the art on the packing coloring, and ts generalization, the $S$-packing coloring. We also list several conjecres and open problems.
COBISS.SI-ID: 23220483
The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three pairwise distinct vertices from $S$ lie on a common geodesic. The $n$-dimensional grid graph $P_\infty^n$ is the Cartesian product of $n$ copies of the two-way infinite path $P_\infty$. It is proved that if $n\in {\mathbb N}$, then ${\rm gp}({P_\infty^n}) = 2^{2^{n-1}}$. The result was earlier known only for $n\in \{1,2\}$ and partially for $n=3$.
COBISS.SI-ID: 32039683