Основные понятия
The paper provides upper and lower bounds on the rectilinear crossing numbers of complete balanced multipartite graphs (Kr
n) and layered graphs (Lr
n).
Аннотация
The paper focuses on the rectilinear crossing numbers of complete balanced multipartite graphs (Kr
n) and layered graphs (Lr
n).
For Kr
n, the authors use a random embedding technique to obtain upper bounds that match the conjectured optimal values up to the leading term. They also provide a lower bound using the rectilinear crossing number of the smaller graph Kr.
For Lr
n, the authors first use the random embedding technique to obtain an upper bound, but then improve upon it by introducing a "planted drawing" technique. This involves using a rectilinear drawing of a smaller graph as a "seed" to construct a larger rectilinear drawing with significantly fewer crossings.
The authors also discuss the relationship between the rectilinear crossing number and the general crossing number for these graph families, and make conjectures about the behavior of these quantities as the number of partitions (r) grows.
Статистика
The paper provides the following key metrics and figures:
The number of crossings in Hill's drawing of Kn is given by the formula H(n) = 1/4 * (n choose 2) * (n-1 choose 2) * (n-2 choose 2) * (n-3 choose 2).
The number of crossings in Harborth's drawing for Kr
n is upper bounded by 1/16 * (r-1)/r * 2 * n^4/4 - 2n^3 + O(n^2).
The expected number of crossings in a random embedding of Kr
n into Hill's drawing of Kn is at most 1/16 * (r-1)/r * 2 * n^4/4 - 3n^3/2 + O(n^2).
The rectilinear crossing number of Kr
n is lower bounded by cr(Kr) * (n/r)^4.
The rectilinear crossing number of Lr
n is upper bounded by (r-1)^2/16r^4 * n^4 + O(n^3).
Цитаты
"Let n ≥r be positive integers. The graph Kr
n, is the complete r-partite graph on n vertices, in which every set of the partition has at least ⌊n/r⌋ vertices. The layered graph, Lr
n, is an r-partite graph on n vertices, in which 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."
"We believe that studying the rectilinear crossing number of Kr
n might shed some light on how optimal rectilinear drawings of Kn look like."