Coloring the vertices of a graph with mutual-visibility property

TL;DR

This paper introduces and studies the mutual-visibility coloring of graphs, exploring its properties, bounds, and algorithms, with particular focus on graphs of diameter two and Cartesian products, advancing understanding of this new coloring concept.

Contribution

It defines the mutual-visibility chromatic number, analyzes its relationship with existing parameters, and provides bounds, algorithms, and specific results for special graph classes.

Findings

Determined the asymptotic growth of the mutual-visibility number for Cartesian products of complete graphs.

Developed a greedy algorithm for mutual-visibility coloring and discussed its efficiency.

Established bounds for the mutual-visibility chromatic number based on graph parameters.

Abstract

Given a graph $G$, a mutual-visibility coloring of $G$ is introduced as follows. We color two vertices $x,y\in V(G)$ with a 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.