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Rectilinear Crossing Numbers of Complete Balanced Multipartite Graphs and Layered Graphs


Core Concepts
The paper provides upper and lower bounds on the rectilinear crossing numbers of complete balanced multipartite graphs (Kr n) and layered graphs (Lr n).
Abstract
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.
Stats
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).
Quotes
"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."

Deeper Inquiries

What other properties of complete balanced multipartite graphs and layered graphs could be explored to gain insights into the structure and crossing number behavior of complete graphs

In addition to the properties explored in the paper, further investigation into the symmetry and connectivity of complete balanced multipartite graphs and layered graphs could provide insights into their structure and crossing number behavior. Symmetry: Analyzing the symmetrical properties of these graphs, such as rotational or reflectional symmetries, could reveal patterns in their crossing numbers. Understanding how symmetries impact the arrangement of edges and vertices in rectilinear drawings may offer clues to optimizing the crossing numbers. Connectivity: Exploring the connectivity between different layers or partitions in layered graphs and complete balanced multipartite graphs could shed light on how the edges are distributed and how crossings occur. Studying the relationships between vertices in different layers and partitions may help in predicting and minimizing the number of crossings. Edge Distribution: Investigating the distribution of edges within and between partitions in complete balanced multipartite graphs could provide valuable information on the crossing behavior. Understanding how edges are distributed unevenly or evenly among partitions may influence the crossing numbers in rectilinear drawings. By delving deeper into these properties, researchers can uncover additional factors that influence the crossing numbers of these graph families and gain a more comprehensive understanding of their structural characteristics.

How might the techniques developed in this paper, such as the random embedding and planted drawing approaches, be applied to study the crossing numbers of other families of graphs

The techniques developed in the paper, such as random embedding and planted drawing approaches, can be applied to study the crossing numbers of various families of graphs beyond complete balanced multipartite graphs and layered graphs. Random Embedding: This technique can be utilized to estimate the crossing numbers of other graph families by mapping vertices randomly onto optimal drawings of specific graphs. By applying this method to different graph structures, researchers can approximate their crossing numbers and identify patterns in the distribution of crossings. Planted Drawings: The concept of planted drawings can be extended to investigate the crossing numbers of diverse graph families. By using specific drawings as seeds and expanding them into larger planted drawings, researchers can explore the optimal arrangements of vertices to minimize crossings in various graph configurations. By adapting these techniques to different graph families and structures, researchers can gain insights into the crossing number behavior of a wide range of graphs and potentially discover new strategies for optimizing rectilinear drawings.

Are there any connections between the rectilinear crossing numbers of these graph families and other graph-theoretic or geometric properties that could be investigated

There are potential connections between the rectilinear crossing numbers of complete balanced multipartite graphs, layered graphs, and other graph-theoretic or geometric properties that warrant further investigation: Planarity: Exploring the relationship between the rectilinear crossing numbers of these graph families and their planarity could reveal interesting connections. Investigating how the planarity of graphs influences their crossing numbers may provide insights into the interplay between geometric properties and crossing behavior. Graph Density: Studying the density of edges in complete balanced multipartite graphs and layered graphs in relation to their crossing numbers could uncover correlations between graph density and crossing complexity. Understanding how the density of edges impacts the occurrence of crossings may lead to new findings in graph theory. Geometric Embeddings: Examining the geometric embeddings of these graph families in different spaces or surfaces could offer insights into their crossing numbers. Investigating how geometric properties affect the optimal rectilinear drawings and crossing numbers may reveal underlying geometric principles governing the arrangement of edges and vertices. By exploring these potential connections, researchers can deepen their understanding of the relationships between rectilinear crossing numbers, graph-theoretic properties, and geometric characteristics, leading to new discoveries in the field of graph theory and combinatorial geometry.
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