20.500.12556/RUL-174362 Coloring the vertices of a graph with mutual-visibility property This 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. graph coloring mutual-visibility set mutual-visibility number mutual-visibility chromatic number graph product diameter two graph barvanje grafa množica vzajemne vidnosti število vzajemne vidnosti kromatično število vzajemne vidnosti produkt grafov graf premera 2 true false true Angleški jezik Slovenski jezik Članek v reviji 2025-10-01 14:27:34 2025-10-01 14:27:40 2025-11-21 03:41:36 0000-00-00 00:00:00 2025 0 0 15 str. iss. 1, article no. 20250193 Vol. 23 Jan. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-01-01 519.17 2391-5455 10.1515/math-2025-0193 251282947 mutual-visibity-coloring.pdf mutual-visibity-coloring.pdf 1 BDC2485A6999682E9F420E83895BA206 8f9a08b1fef4bc1d97ecb4019539526bbf2c60b16f709b264bf69b86c009e0c6 ff51144c-9ec5-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=219160 https://www.degruyterbrill.com/document/doi/10.1515/math-2025-0193/html 1 d1c66d5e-9ec1-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=219157 Fakulteta za matematiko in fiziko 0 0 0