Sharp threshold for embedding balanced spanning trees in random geometric graphs

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-05-09 DOI:10.1002/jgt.23106

Alberto Espuny Díaz, Lyuben Lichev, Dieter Mitsche, Alexandra Wesolek

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Abstract

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph $G (n, r, d)$ . In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

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在随机几何图中嵌入平衡生成树的锐阈值

如果一个顶点的度数只取决于它到根的距离，那么这棵有根树就是平衡的。在本文中，我们确定了在随机几何图形中出现平衡生成树大家族的尖锐阈值。特别是，我们找到了平衡二叉树的尖锐阈值。更一般地说，我们证明了所有度数均匀有界且高度趋于无穷大的平衡树序列都会出现在一个尖锐阈值之上，而没有一个序列会出现在同一值之下。我们的结果更普遍地适用于满足顶点集分布的温和条件的几何图形，我们还提供了一种多项式时间算法来寻找这样的树。

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

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .