Coloring, location and domination of corona graphs

A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those structures of a graph satisfying some domination property together with other conditions on the vertices of $G$. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-$k$ colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, throughout some relationships between the distance-$k$ chromatic number of corona graphs and the distance-$k$ chromatic number of its factors. Moreover, we give the exact value of the distance-$k$ chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating-domination number of corona graphs. We give closed formulaes for the $k$-domination number, the distance-$k$ domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page

Total mutual-visibility in graphs with emphasis on lexicographic and Cartesian products

Given a connected graph $G$, the total mutual-visibility number of $G$, denoted $\mu_t(G)$, is the cardinality of a largest set $S\subseteq V(G)$ such that for every pair of vertices $x,y\in V(G)$ there is a shortest $x,y$-path whose interior vertices are not contained in $S$. Several combinatorial properties, including bounds and closed formulae, for $\mu_t(G)$ are given in this article. Specifically, we give several bounds for $\mu_t(G)$ in terms of the diameter, order and/or connected domination number of $G$ and show characterizations of the graphs achieving the limit values of some of these bounds. We also consider those vertices of a graph $G$ that either belong to every total mutual-visibility set of $G$ or does not belong to any of such sets, and deduce some consequences of these results. We determine the exact value of the total mutual-visibility number of lexicographic products in terms of the orders of the factors, and the total mutual-visibility number of the first factor in the product. Finally, we give some bounds and closed formulae for the total mutual-visibility number of Cartesian product graphs