Crossing Numbers and Combinatorial Characterization of Monotone Drawings of $$K_n$$Kn

@article{Balko2015CrossingNA,
  title={Crossing Numbers and Combinatorial Characterization of Monotone Drawings of \$\$K\_n\$\$Kn},
  author={Martin Balko and Radoslav Fulek and Jan Kyn{\vc}l},
  journal={Discrete \& Computational Geometry},
  year={2015},
  volume={53},
  pages={107-143}
}
In 1958, Hill conjectured that the minimum number of crossings in a drawing of $$K_n$$Kn is exactly $$Z(n) = \frac{1}{4} \big \lfloor \frac{n}{2}\big \rfloor \big \lfloor \frac{n-1}{2}\big \rfloor \big \lfloor \frac{n-2}{2}\big \rfloor \big \lfloor \frac{n-3}{2}\big \rfloor $$Z(n)=14⌊n2⌋⌊n-12⌋⌊n-22⌋⌊n-32⌋. Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $$x$$x-monotone curves. In fact, our proof… 
The Crossing Number of Seq-Shellable Drawings of Complete Graphs
TLDR
The class of seq-shellable drawings is introduced and it is shown that bishellability impliesseq-shellability and exhibit a non-bishellable but seq- shellable drawing of $K_{11}$, therefore the class ofseq-Shellable drawings strictly contains theclass of bishellable drawings.
Bishellable drawings of $K_n$
The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is $ H(n) = \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut
Bishellable drawings of Kn
TLDR
The main result is that $(\lfloor \frac{n}{2} \rfloor\!-\!2)-bishellability also guarantees, with a simpler proof than for $s-shellability, that a drawing has at least $H(n)$ crossings.
The Crossing Number of Single-Pair-Seq-Shellable Drawings of Complete Graphs
TLDR
The Harary-Hill Conjecture is proved for a new class of single-pair-seq-shellable drawings and the notion of $k$-deviations is introduced as the difference between an optimal and the actual number of k-edges, and the necessity of a fixed reference face is relaxed.
Closing in on Hill's Conjecture
TLDR
It is shown that Hill's conjecture is "asymptotically at least 98.5% true", the best known asymptotic lower bound for the crossing number of complete bipartite graphs.
Edge-Minimum Saturated k-Planar Drawings
TLDR
A generic framework is established to determine the minimum number of edges among all $n-vertex saturated $k$-planar drawings in many natural classes.
The Crossing Tverberg Theorem
TLDR
A strengthening of Tverberg's theorem is proved that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections.
Simple Realizability of Complete Abstract Topological Graphs Simplified
TLDR
This work characterize simply realizable complete AT-graphs by a finite set of forbidden AT-subgraphs, each with at most six vertices, which implies a straightforward polynomial algorithm for testing simple realizability of complete AT -graphs.
On Disjoint Holes in Point Sets
TLDR
Using computer assistance, the program can be used to verify that every set of 17 points contains a 6-gon within significantly smaller computation time than the original program by Szekeres and Peters (2006).
...
...

References

SHOWING 1-10 OF 63 REFERENCES
Shellable Drawings and the Cylindrical Crossing Number of $$K_{n}$$Kn
TLDR
The techniques developed provide a unified proof of the Harary–Hill conjecture for shellable drawings and it is proved that all cylindrical, x-bounded, monotone, and 2-page drawings of K_n Kn are s-shellable for some s≥n/2 and thus they all have at least Z(n) = 14n2n-12n-22n-32 crossings.
New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn
TLDR
The proof is based on a result about the structure of sets attaining the rectilinear crossing number, for which the convex hull is always a triangle and provides improved upper bounds on the maximum number of halving edges a point set can have.
On $(\le k)$-edges, crossings, and halving lines of geometric drawings of $K_n$
Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is
Reduction of symmetric semidefinite programs using the regular $$\ast$$-representation
TLDR
A general technique to reduce the size ofSemidefinite programming problems on which a permutation group is acting is described, based on a low-order matrix based on the representation of the commutant (centralizer ring) of the matrix algebra generated by the permutation matrices.
Note on the Pair-crossing Number and the Odd-crossing Number
TLDR
It is shown that if the pair-crossing number of G is k, then its crossing number is at most O(k2/log 2k).
On ≤k-Edges, Crossings, and Halving Lines of Geometric Drawings of Kn
TLDR
It is shown that the previously best known lower bound on E≤k(n) is tight for k<⌈(4n−2)/9⌉ and improve it for all k≥⌊( 4n− 2)/9 ⌉.
The 2-page crossing number of Kn
TLDR
A novel and innovative technique is developed to investigate crossings in drawings of K<sub>n</sub, and a powerful theorem is extended that expresses the number of crossings in a rectilinear drawing of K(sub)n in terms of its number of k-edges to the topological setting.
More on the crossing number of Kn: Monotone drawings
Simple Realizability of Complete Abstract Topological Graphs in P
  • J. Kynčl
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 2011
TLDR
This work presents a polynomial algorithm which decides whether a given complete AT- graph is simply realizable and shows that other similar realizability problems for (complete) AT-graphs are NP-hard.
Towards a Theory of Geometric Graphs
On the complexity of the linkage reconfiguration problem by H. Alt, C. Knauer, G. Rote, and S. Whitesides Falconer conjecture, spherical averages and discrete analogs by G. Arutyunyants and A.
...
...