Up to a double cover, every regular connected graph is isomorphic to a Schreier graph

IF 0.4 4区数学 Q4 MATHEMATICS

Bulletin of the Belgian Mathematical Society-Simon Stevin Pub Date : 2022-03-01 DOI:10.36045/j.bbms.210416

P. Leemann

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引用次数: 1

Abstract

We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph. The aim of this short note is to provide a proof of the following result. Proposition 1. Let G be a d-regular connected graph. Then either G is isomorphic to a Schreier graph or G has a double-cover H which is isomorphic to a Schreier graph. While we were not able to find a reference to the above result in the literature, we do not claim any priority on it. In fact, this note was inspired by the following remark, which can be found at the end of Section 7 of [4]: “In fact up to the cover of degree 2 any regular graph can be realized as a Schreier graph [8]”. However, [8] seems to treat only the case of graphs of even degree. Section 1 contains all the definitions as well as a discussion on the unusual concept of degenerated loop. The short Section 2 contains more materials on coverings and perfect matchings, as well as the proof of Proposition 1. Acknowledgements The author is thankful to A. Georgakopoulos, R. Grigorchuk and M. de la Salle for comments on a previous version of this note. The author was supported by NSF Grant No. 200021_188578.

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在双盖之前，每一个正则连通图都与Schreier图同构

证明了每一个连通的局部有限正则图都有一个与Schreier图同构的双盖。这篇短文的目的是为以下结果提供证明。命题1。设G是一个d正则连通图。要么G与Schreier图同构要么G有一个双盖H与Schreier图同构。虽然我们无法在文献中找到对上述结果的参考，但我们不认为它具有任何优先权。实际上，这篇笔记的灵感来自于文献[4]第7节末尾的一句话:“事实上，直到2度的覆盖，任何正则图都可以被实现为Schreier图[8]”。然而，[8]似乎只处理偶数次图的情况。第1节包含了所有的定义，并讨论了退化环路的不同寻常的概念。较短的第2节包含更多关于覆盖物和完美匹配的材料，以及命题1的证明。作者感谢a . Georgakopoulos、R. Grigorchuk和M. de la Salle对本文前一版本的评论。作者获得国家自然科学基金资助(200021_188578)。

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来源期刊

Bulletin of the Belgian Mathematical Society-Simon Stevin 数学-数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The Bulletin of the Belgian Mathematical Society - Simon Stevin (BBMS) is a peer-reviewed journal devoted to recent developments in all areas in pure and applied mathematics. It is published as one yearly volume, containing five issues. The main focus lies on high level original research papers. They should aim to a broader mathematical audience in the sense that a well-written introduction is attractive to mathematicians outside the circle of experts in the subject, bringing motivation, background information, history and philosophy. The content has to be substantial enough: short one-small-result papers will not be taken into account in general, unless there are some particular arguments motivating publication, like an original point of view, a new short proof of a famous result etc. The BBMS also publishes expository papers that bring the state of the art of a current mainstream topic in mathematics. Here it is even more important that at leat a substantial part of the paper is accessible to a broader audience of mathematicians. The BBMS publishes papers in English, Dutch, French and German. All papers should have an abstract in English.