On the rectilinear crossing number of complete balanced multipartite graphs and balanced 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 and its vertices are points in general position. 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 balanced r-partite graph on n vertices, in which every set of the partition has at least $\lfloor n/r\rfloor$ vertices. The balanced layered graph, $L_n^r$, is an r-partite graph on n vertices, where n is multiple of r. Every partition of $L_n^r$ contains $n/r$ vertices; 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, and these are the only edges of $L_n^r$. In this paper, we give upper bounds on the rectilinear crossing numbers of $K_n^r$ and $L_n^r$.