The circular $k$-partite crossing number of $K_{m,n}$

On the outerplanar crossing numbers of complete multipartite graphs

We calculate the outerplanar crossing numbers of complete multipartite graphs which have $n$ partite sets with $m$ vertices and one partite set with $p$ vertices, where either $p|mn$ or $mn|p$.

Rectilinear Crossings in Complete Balanced d-Partite d-Uniform Hypergraphs

In this paper, we study the embedding of a complete balanced $d$-partite $d$-uniform hypergraph with all its $nd$ vertices represented as points in general position in $\mathbb{R}^d$ and each hyperedge drawn as a convex hull of $d$ corresponding vertices. We assume that the set of vertices is partitioned into $d$ disjoint sets, each of size $n$, such that each of the vertices in a hyperedge is from a different set. Two hyperedges are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors. Using the Generalized Colored Tverberg Theorem, we observe…

Crossing numbers of complete tripartite and balanced complete multipartite graphs

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr'(K_{n_1,n_2})\le Z(n_1,n_2):= n_1/2*(n_1-1)/2*n_2/2*(n_2-1)/2. We define an analogous bound A(n_1,n_2,n_3) for the complete tripartite graph K_{n_1,n_2,n_3}, and prove that cr'(K_{n_1,n_2,n_3})\le A({n_1,n_2,n_3}). We also show that for n large enough, 0.973 A(n,n,n) \le cr'(K_{n…

The crossing number of the complete 4-partite graph $K_{1,1,m,n}$

Let $\textrm{cr}(G)$ denote the crossing number of a graph $G$. The well-known Zarankiewicz's conjecture (ZC) asserted $\textrm{cr}(K_{m,n})$ in 1954. In 1971, Harborth gave a conjecture (HC) on $\textrm{cr}(K_{x_1,...,x_n})$. HC on $K_{1,m,n}$ is verified if ZC is true by Ho et al. in 2021. In this paper, we showed the following results: If both $m$ and $n$ are even, then \[\textrm{cr}(K_{1,1,m,n})\geq \frac{1}{2}(\textrm{cr}(K_{m+1,n+3})+\textrm{cr}(K_{m+3,n+1})-mn-\frac{1}{4}(m^2+n^2));\] If both $m$ and $n$ are odd, then \[\textrm{cr}(K_{1,1,m,n})\geq \frac{1}{2}(\textrm{cr}(K_{1,m+1,n+1})…

关于完全平衡多部图和分层图的直线交叉数

On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs

本文研究完全平衡 r 部图 K_n^r 与相邻层全连接的分层图 L_n^r，通过构造直线画法给出二者直线交叉数的上界。

Treewidth, Circle Graphs and Circular Drawings

A circle graph is an intersection graph of a set of chords of a circle. We describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects'. Our results imply that treewidth and Hadwiger number are linearly tied on the class of circle graphs, and that the unavoidable induced subgraphs of a vertex-minor-closed class with large treewidth are the usual suspects if and only if the class has bounded rank-width. Using the same tools, we also study the treewidth of graphs $G$ that have a circular drawing whose crossing gra…