Coloring the vertices of a graph with mutual-visibility property
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909280588791808 |
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| author | Klavžar, Sandi Kuziak, Dorota Tripodoro, Juan Carlos Valenzuela Yero, Ismael G. |
| author_facet | Klavžar, Sandi Kuziak, Dorota Tripodoro, Juan Carlos Valenzuela Yero, Ismael G. |
| contents | 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 $χ_μ(G)$. Relationships between $χ_μ(G)$ 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 $G$ for which $χ_μ(G)=2$ are also considered. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_03132 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coloring the vertices of a graph with mutual-visibility property Klavžar, Sandi Kuziak, Dorota Tripodoro, Juan Carlos Valenzuela Yero, Ismael G. Combinatorics 05C12, 05C15, 05C69 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 $χ_μ(G)$. Relationships between $χ_μ(G)$ 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 $G$ for which $χ_μ(G)=2$ are also considered. |
| title | Coloring the vertices of a graph with mutual-visibility property |
| topic | Combinatorics 05C12, 05C15, 05C69 |
| url | https://arxiv.org/abs/2408.03132 |