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~$L_n^r$.

翻译：图的直线绘制是指将图的边表示为平面中直线段的绘制方式。图的直线交叉数是指在其所有直线绘制中，边对交叉的最小数量。设$n \ge r$为正整数。图$K_n^r$是包含$n$个顶点的完全$r$部图，其中每个划分集合至少包含$\lfloor n/r \rfloor$个顶点。分层图$L_n^r$是包含$n$个顶点的$r$部图，对于每个$1\le i \le r-1$，第$i$个划分中的所有顶点均与第$i+1$个划分中的所有顶点相邻。本文给出了$K_n^r$和$L_n^r$的直线交叉数的上界。