The Maximum Rectilinear Crossing Number of the n Dimensional Cube Graph

We find a.nd prove the maximum rectilinear crossing n1.1mber of the three-dimensional cube graph (Q3). We demonstrate a method of drawing then-cube graph, Qn., with many crossings, and thus find a lower bound for the maximum rectilinear crossing number of Qn. We conjecture that this bound is sharp. We also prove an upper bound for the maximum rectilinear crossing number of Qn

Maximum Rectilinear Crossing Number of Uniform Hypergraphs

We improve the lower bound on the $d$-dimensional rectilinear crossing number of the complete $d$-uniform hypergraph having $2d$ vertices to $\Omega(2^d d)$ from $\Omega(2^d \sqrt{d})$. We also establish that $3$-dimensional rectilinear crossing number of a complete $3$-uniform hypergraph having $n \geq 9$ vertices is at least $\dfrac{43}{42}\dbinom{n}{6}$. Anshu et al. conjectured that among all $d$-dimensional convex drawings of a complete $d$-uniform hypergraph having $n$ vertices, the number of crossing pairs of hyperedges is maximized if all its vertices are placed on the $d$-dimensional moment curve and proved this conjecture for $d=3$. It is trivially true for $d = 2$, since any convex drawing of the complete graph $K_n$ produces $n \choose 4$ pairs of crossing edges. We prove that their conjecture is true for $d=4$ by using Gale transform. In fact, we prove a stronger statement. We prove that among all $4$-dimensional rectilinear drawings of a complete $4$-uniform hypergraph having $n$ vertices, the number of crossing pairs of hyperedges is maximized if all its vertices are placed on the $4$-dimensional moment curve. We also prove that the maximum $d$-dimensional rectilinear crossing number of a complete $d$-partite $d$-uniform balanced hypergraph is $(2^{d-1}-1){n \choose 2}^d$, where $n$ denotes the number of vertices in each part. We then prove that finding the maximum $d$-dimensional rectilinear crossing number of an arbitrary $d$-uniform hypergraph is NP-hard and give a randomized scheme to create a $d$-dimensional rectilinear drawing of a $d$-uniform hypergraph $H$ producing the number of crossing pairs of hyperedges up to a constant factor of the maximum $d$-dimensional rectilinear crossing number of $H$.Comment: 23 pages, 3 figures. The paper also contains computer aided proof. To view the results obtained by computers, please mail the author

The Maximum Rectilinear Crossing Number of the Wheel Graph

We find and prove the maximum rectilinear crossing number of the wheel graph. First, we illustrate a picture of the wheel graph with many crossings to prove a lower bound. We then prove that this bound is sharp. The treatment is divided into two cases for n even and n odd

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi: V \to P$ such that the induced straight-line drawings of $G_1,\dots,G_k$ under $\phi$ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for $\exists\mathbb{R}$, the existential theory of the reals. Hence the problem $k$-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to $k$-SGE, with the property that both numbers $k$ and $n$ are linear in the number of pseudolines. This implies not only the $\exists\mathbb{R}$-hardness result, but also a $2^{2^{\Omega (n)}}$ lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only $2^{2^{\Omega (\sqrt{n})}}$