On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915096127602688 |
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| author | Fabila-Monroy, Ruy Paul, Rosna Viafara-Chanchi, Jenifer Weinberger, Alexandra |
| author_facet | Fabila-Monroy, Ruy Paul, Rosna Viafara-Chanchi, Jenifer Weinberger, Alexandra |
| contents | A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let $n \ge r$ be positive integers. The graph $K_n^r$, is the complete $r$-partite graph on $n$ vertices, in which every set of the partition has at least $\lfloor n/r \rfloor$ vertices. The layered graph, $L_n^r$, is an $r$-partite graph on $n$ vertices, in which for every $1\le i \le r-1$, all the vertices in the $i$-th partition are adjacent to all the vertices in the $(i+1)$-th partition. In this paper, we give upper bounds on the rectilinear crossing numbers of $K_n^r$ and~$L_n^r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13155 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs Fabila-Monroy, Ruy Paul, Rosna Viafara-Chanchi, Jenifer Weinberger, Alexandra Combinatorics Computational Geometry A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let $n \ge r$ be positive integers. The graph $K_n^r$, is the complete $r$-partite graph on $n$ vertices, in which every set of the partition has at least $\lfloor n/r \rfloor$ vertices. The layered graph, $L_n^r$, is an $r$-partite graph on $n$ vertices, in which for every $1\le i \le r-1$, all the vertices in the $i$-th partition are adjacent to all the vertices in the $(i+1)$-th partition. In this paper, we give upper bounds on the rectilinear crossing numbers of $K_n^r$ and~$L_n^r$. |
| title | On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs |
| topic | Combinatorics Computational Geometry |
| url | https://arxiv.org/abs/2404.13155 |