Coloring the vertices of a graph with mutual-visibility property

Klavžar, Sandi (author), Kuziak, Dorota (author), Valenzuela Tripodoro, Juan Carlos (author), Yero, Ismael G. (author)

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.

• Language: English
• Keywords: graph coloring, mutual-visibility set, mutual-visibility number, mutual-visibility chromatic number, graph product, diameter two graph
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2025
• Year: 2025
• Number of pages: 15 str.
• Numbering: Vol. 23, iss. 1, article no. 20250193
• PID: 20.500.12556/RUL-174362
• UDC: 519.17
• ISSN on article: 2391-5455
• DOI: 10.1515/math-2025-0193
• COBISS.SI-ID: 251282947
• Publication date in RUL: 01.10.2025

Record is a part of a journal

• Title: Open Mathematics
• Shortened title: Open Math.
• Publisher: De Gruyter Open
• ISSN: 2391-5455
• COBISS.SI-ID: 17824345

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: barvanje grafa, množica vzajemne vidnosti, število vzajemne vidnosti, kromatično število vzajemne vidnosti, produkt grafov, graf premera 2