Faces in rectilinear drawings of complete graphs

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics Pub Date : 2025-07-11 DOI:10.1016/j.ejc.2025.104217

Martin Balko , Anna Brötzner , Fabian Klute , Josef Tkadlec

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引用次数: 0

Abstract

We initiate the study of extremal problems about faces in convex rectilinear drawings of

K_{n}

, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of

K_{n}

does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex

k

-gon with

k \geq 6

A convex rectilinear drawing of

K_{n}

is regular if its vertices correspond to vertices of a regular convex

n

-gon. We characterize positive integers

n

for which regular drawings of

K_{n}

contain a face forming a convex 5-gon.

To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.

查看原文本刊更多论文

完全图的直线图中的面

我们开始研究Kn的凸直线图中关于面的极值问题，即顶点由平面上凸位置的点表示，边缘由代表端点的点之间的线段表示的图。我们证明，如果Kn的凸直线图不包含至少三条边的公共内点，则总有一个面形成凸5-gon，而存在这样的图，没有任何面形成k≥6的凸k-gon。如果一个Kn的凸直线图的顶点对应于一个正则凸n-gon的顶点，那么它就是正则的。我们描述正整数n，其中Kn的正则图包含一个形成凸5-gon的面。据我们所知，这类问题在以前的文献中没有被考虑过，所以我们也提出了几个新的自然开放问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

European Journal of Combinatorics 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.