On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs
<|reference_start|>On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs: 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~$L_n^r$.<|reference_end|>
@article{fabila-monroy2024on,
    title={On the rectilinear crossing number of complete balanced multipartite
  graphs and layered graphs},
    author={Ruy Fabila-Monroy and Rosna Paul and Jenifer Viafara-Chanchi and
  Alexandra Weinberger},
    journal={arXiv preprint arXiv:2404.13155},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.13155},
    primaryClass={math.CO cs.CG}
}