Coloring the vertices of a graph with mutual-visibility property

Given a graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G , a mutual-visibility coloring of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G is a coloring of the vertices of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G satisfying the following. Two vertices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> x,y∈ V(G) can be colored with the same color, if there is a shortest <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:math> x,y -path whose internal vertices have different colors than <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> </m:math> x and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>y</m:mi> </m:math> y . The smallest number of colors among all mutual-visibility colorings of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G is the mutual-visibility chromatic number of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G , which is denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>χ</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {χ }μ(G) . Relationships between <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>χ</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {χ }μ(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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G for which <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>χ</m:mi> </m:mrow> <m:mrow> <m:mi>μ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:math> {χ }μ(G)=2 are also considered.