Coloring the vertices of a graph with mutual-visibility propertyKlavžar, Sandi (Avtor) Kuziak, Dorota (Avtor) Valenzuela Tripodoro, Juan Carlos (Avtor) Yero, Ismael G. (Avtor) graph coloringmutual-visibility setmutual-visibility numbermutual-visibility chromatic numbergraph productdiameter two graphThis article combines two contrasted graph theory topics. In one hand, the notion of coloring the vertices of a graph satisfying certain properties, which is a classical area in graph theory. In the second one, the notion of mutual-visibility between pairs of vertices, which is yet a fresh topic, but already an established and hot one due to several connections with other classical combinatorial topics. Given a graph $G$, a mutual-visibility coloring of $G$ is introduced as follows. We color two vertices $x,y\in V(G)$ with the same color, if there is a shortest $x,y$-path whose internal vertices have different colors than $x,y$. The smallest number of colors needed in a mutual-visibility coloring of $G$ is the mutual-visibility chromatic number of $G$, which is denoted $\chi_{\mu}(G)$. Relationships between $\chi_{\mu}(G)$ 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 $G$ for which $\chi_{\mu}(G)=2$ are also considered.20252025-10-01 14:22:28Neznano23770UDK: 519.17ISSN pri članku: 2391-5455DOI: 10.1515/math-2025-0193COBISS_ID: 251282947sl