Matching Book Thickness of Halin Graphs

完全图$K_n$的星荫度松弛书厚度

Star arboricity relaxed book thickness of $K_n$

该研究针对完全图$K_n$的书嵌入问题，提出通过添加一个交叉帽页或两个普通页，可将页面子图全部选为星森林，优化了书嵌入的页面子图结构。

关于若干循环图的可分散性

On dispersability of some circulant graphs

该研究围绕图的匹配书厚度与可分散性展开，针对环的Harary立方体中的二部及多数非二部循环图，还有多种更高次幂循环图，给出了最小页数的匹配书嵌入结果。

Four Pages Are Indeed Necessary for Planar Graphs

An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this pro…

两个图的F-和的匹配书嵌入

The matching book embedding of the $F$-sum of two graphs

本文研究了外平面图的可分散性，并确定了任意简单图与可分散双分图的F-和的匹配书厚度上界。

On Dispersable Book Embeddings

In a dispersable book embedding, the vertices of a given graph $G$ must be ordered along a line l, called spine, and the edges of G must be drawn at different half-planes bounded by l, called pages of the book, such that: (i) no two edges of the same page cross, and (ii) the graphs induced by the edges of each page are 1-regular. The minimum number of pages needed by any dispersable book embedding of $G$ is referred to as the dispersable book thickness $dbt(G)$ of $G$. Graph $G$ is called dispersable if $dbt(G) = Δ(G)$ holds (note that $Δ(G) \leq dbt(G)$ always holds). Back in 1979, Bernhart a…

Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\…

Bipartite cubic planar graphs are dispersable

The book embedding of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. A book embedding is matching if the vertices in the same page have maximum degree at most 1. The matching book thickness is the minimum number of pages in which a graph can be matching book embedded. A graph $G$ is dispersable if and only if $mbt(G)=Δ(G)$. In this paper, we prove that bipartite cubic planar graphs are dispersable.

关于奇环面网格近乎可分散性的一个注记

A note on the Nearly Dispersability of Odd Toroidal Grids

本文在作者先前确定奇环面网格近乎可分散性结果的基础上，进一步为该结果提供了一个简要的证明。

On dispersability of some products of cycles

We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs is five if at least one of the cycles has length 3 or 5.

Geometric Thickness of Complete Graphs

We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantities, the (graph-theoretical) thickness and the book thickness. We investigate the geometric thickness of the family of complete graphs, K_n. We show that the geometric thickness of K_n lies between ceiling((n/5.646) + 0.342) and ceiling(n/4), and we give exact values of the geometric thickness of K_n fo…

Book Embeddings of Graph Products

A $k$-stack layout (also called a $k$-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing edges with respect to the vertex order. The stack number (book thickness, page number) of a graph is the minimum $k$ such that it admits a $k$-stack layout. A $k$-queue layout is defined similarly, except that no two edges in a single set may be nested. It was recently proved that graphs of various non-minor-closed classes are subgraphs of the strong product of a path and a graph with bounded treewidth. Motivated by this dec…