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 >= 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 & LeftFloor;n/r & RightFloor; 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 <= i <= 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).

Every simple drawing of a graph in the plane naturally induces a rotation system, but it is easy to exhibit a rotation system that does not arise from a simple drawing in the plane. We extend this to all surfaces: for every fixed surface $\Sigma$, there is a rotation system that does not arise from a simple drawing in $\Sigma$.

We consider bichromatic point sets with n red and n blue points and study straight-line bichromatic perfect matchings on them. We show that every such point set in convex position admits a matching with at least 3n^2/8-n/2+c crossings, for some -1/2≤ c ≤1/8 . This bound is tight since for any k> 3n^2/8 -n/2+1/8 there exist bichromatic point sets that do not admit any perfect matching with k crossings.

Let $S$ be a planar point set in general position, and let $\mathcal{P}(S)$ be the set of all plane straight-line paths with vertex set $S$. A flip on a path $P \in \mathcal{P}(S)$ is the operation of replacing an edge $e$ of $P$ with another edge $f$ on $S$ to obtain a new valid path from $\mathcal{P}(S)$. It is a long-standing open question whether for every given point set $S$, every path from $\mathcal{P}(S)$ can be transformed into any other path from $\mathcal{P}(S)$ by a sequence of flips. To achieve a better understanding of this question, we show that it is sufficient to prove the statement for plane spanning paths whose first edge is fixed. Furthermore, we provide positive answers for special classes of point sets, namely, for wheel sets and generalized double circles (which include, e.g., double chains and double circles).

For a simple drawing D of the complete graph $$K_n$$ , two (plane) subdrawings are compatible if their union is plane. Let $$\mathcal {T}_D$$ be the set of all plane spanning trees on D and $$\mathcal {F}(\mathcal {T}_D)$$ be the compatibility graph that has a vertex for each element in $$\mathcal {T}_D$$ and two vertices are adjacent if and only if the corresponding trees are compatible. We show, on the one hand, that $$\mathcal {F}(\mathcal {T}_D)$$ is connected if D is a cylindrical, monotone, or strongly c-monotone drawing. On the other hand, we show that the subgraph of $$\mathcal {F}(\mathcal {T}_D)$$ induced by stars, double stars, and twin stars is also connected. In all cases the diameter of the corresponding compatibility graph is at most linear in n.

For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least C_n/2 different plane perfect matchings, where C_n/2 is the n /2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k≤1/64n^2-35/32n√(n)+1225/64n , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most 5/72n^2-n/4 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n . (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k=0,1,2 , and maximize the number of perfect matchings with ( [ n/2; 2 ]) crossings and with ( [ n/2; 2 ]) -1 crossings.

For sets of $$n = 2m$$ points in general position in the plane we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least $$C_m$$ different plane perfect matchings, where $$C_m$$ is the m-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every $$k\le \frac{1}{64}n^2-O(n \sqrt{n})$$ , any set of n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has fewer than $$\frac{5}{72}n^2$$ crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for $$k=0,1,2$$ , and maximize the number of perfect matchings with $$\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) $$ crossings and with $${\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) }\!-\!1$$ crossings.

Recently, the second and third author showed that complete geometric graphs on 2n vertices in general cannot be partitioned into n plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

Károlyi, Pach, and Tóth proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two partial results towards such a generalization. First, we show that the statement holds for cylindrical simple drawings. (In a cylindrical drawing, all vertices are placed on two concentric circles and no edge crosses either circle.) Second, we introduce a relaxation of the problem in which the graph is k-edge-colored, and the target structure must be hypochromatic, that is, avoid (at least) one color class. In this setting, we show that every ⌈ (n+5)/6⌉ -edge-colored monotone simple drawing of K_n contains a hypochromatic plane spanning tree. (In a monotone drawing, every edge is represented as an x-monotone curve.)