Coloring the vertices of a graph with mutual-visibility property

Abstract Given a graph G G, a mutual-visibility coloring of G G is a coloring of the vertices of G G satisfying the following. Two vertices x, y ∈ V (G) x, y V (G) can be colored with the same color, if there is a shortest x, y x, y -path whose internal vertices have different colors than x x and y y. The smallest number of colors among all mutual-visibility colorings of G G is the mutual-visibility chromatic number of G G, which is denoted by χ μ (G) (G). Relationships between χ μ (G) (G) and its two parent ones, the chromatic number and the mutual-visibility number, are presented. Graphs of diameter two are considered, and in particular, the asymptotic growth of the mutual-visibility number of the Cartesian product of complete graphs is determined. A greedy algorithm that finds a mutual-visibility coloring is designed, and several possible scenarios on its efficiency are discussed. Several bounds are given in terms of other graph parameters such as the diameter, the order, the maximum degree, the degree of regularity of regular graphs, and/or the mutual-visibility number. For the corona products, it is proved that the value of its mutual-visibility chromatic number depends on that of the first factor of the product. Graphs G G for which χ μ (G) = 2 (G) =2 are also considered.