树的笛卡尔积的λ4连通性

Hengzhe Li, Jiajia Wang, Rongxia Hao
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引用次数: 0

摘要

给定一个连通图[公式:见文]和[公式:见文]与[公式:见文]的连通图[公式:见文],一个[公式:见文]树就是[公式:见文]的这样一个子图[公式:见文],它是一个有[公式:见文]的树。如果[公式:见文本],两个[公式:见文本]-树[公式:见文本]和[公式:见文本]是边不相交的。设[公式:见文]为[公式:见文]中一组边不相交的[公式:见文]树的最大大小。[公式:见文]的[公式:见文]-连通性被定义为[公式:见文]。在本文中,我们首先用范引理和König-ore公式证明了边不相交树的一些结构性质。然后,确定了树的笛卡尔积的连通性。也就是说,设[Formula: see text]为树,如果[Formula: see text]为每个[Formula: see text],则[Formula: see text]为[Formula: see text],否则为[Formula: see text]。作为推论,[公式:见文]-连通性的一些图类,如超立方体和网格可以直接得到。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The λ4-Connectivity of the Cartesian Product of Trees
Given a connected graph [Formula: see text] and [Formula: see text] with [Formula: see text], an [Formula: see text]-tree is a such subgraph [Formula: see text] of [Formula: see text] that is a tree with [Formula: see text]. Two [Formula: see text]-trees [Formula: see text] and [Formula: see text] are edge-disjoint if [Formula: see text]. Let [Formula: see text] be the maximum size of a set of edge-disjoint [Formula: see text]-trees in [Formula: see text]. The [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. In this paper, we first show some structural properties of edge-disjoint [Formula: see text]-trees by Fan Lemma and König-ore Formula. Then, the [Formula: see text]-connectivity of the Cartesian product of trees is determined. That is, let [Formula: see text] be trees, then [Formula: see text] if [Formula: see text] for each [Formula: see text], otherwise [Formula: see text]. As corollaries, [Formula: see text]-connectivity for some graph classes such as hypercubes and meshes can be obtained directly.
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