On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs (2404.13155v1)
Abstract: 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 $K_nr$, 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, $L_nr$, 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 $K_nr$ and~$L_nr$.
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