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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.12323 |
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| _version_ | 1866909518770733056 |
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| author | Díaz, Josep Diner, Öznur Yaşar Serna, Maria Serra, Oriol |
| author_facet | Díaz, Josep Diner, Öznur Yaşar Serna, Maria Serra, Oriol |
| contents | An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color.
In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from $h\ge 2$ independent random geometric graphs $G(n,r)$ on the unit square. We define an edge-coloring by coloring the edges according to the copy of $G(n,r)$ they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that $r(n)=\left(\frac{\log n}{n}\right)^{\frac{h-1}{2h}}$ is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erdős-R\' enyi random model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12323 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rainbow connectivity of multilayered random geometric graphs Díaz, Josep Diner, Öznur Yaşar Serna, Maria Serra, Oriol Combinatorics 05C80, 05C82, 68R10 An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from $h\ge 2$ independent random geometric graphs $G(n,r)$ on the unit square. We define an edge-coloring by coloring the edges according to the copy of $G(n,r)$ they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that $r(n)=\left(\frac{\log n}{n}\right)^{\frac{h-1}{2h}}$ is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erdős-R\' enyi random model. |
| title | Rainbow connectivity of multilayered random geometric graphs |
| topic | Combinatorics 05C80, 05C82, 68R10 |
| url | https://arxiv.org/abs/2407.12323 |