标题： 完全图的书本穿越数和小局部凸穿越数 显示英文标题

标题： Book crossing numbers of the complete graph and small local convex crossing numbers

摘要： 一个$ k $页的图$ G $的书式绘制是在具有公共边界$ l $（一条线）的$ k $个半平面上绘制$ G $，其中顶点位于$ l $上且边不能穿过$ l $。 The $ k $-page book crossing number of the graph $ G $, denoted by $ \nu_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. 我们改进了所有$ k\geq 14 $下的$ \nu_k(K_n) $的下界，并在$ 2 < n/k \leq 3 $成立时确定了$ \nu_k(K_n) $。 我们的证明依赖于对局部交叉数较小的凸图中边的数量进行界限定理。 特别是，我们确定了对于$ \ell\leq 4 $，局部交叉数最多为$ \ell $的凸图所能拥有的最大边数。 显示英文摘要

摘要： A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k $-page book crossing number of the graph $ G $, denoted by $ \nu_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ \nu_k(K_n) $ for all $ k\geq 14 $ and determine $ \nu_k(K_n) $ whenever $ 2 < n/k \leq 3 $. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a convex graph with local crossing number at most $ \ell $ can have for $ \ell\leq 4 $.