关于扎哈尔和厄尔多斯的一个问题

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-12-11 DOI:10.1016/j.jctb.2023.11.004

Tung Nguyen , Alex Scott , Paul Seymour

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引用次数: 0

摘要

如果图 G 的两个子图 A,B 的顶点不相交，并且没有连接它们的边，那么这两个子图就是反完全子图。如果 G 是一个具有有界簇数和足够大色度数的图，那么它是否真的有两个都具有大色度数的反完全子图？这是 El-Zahar 和 Erdős 在 1986 年提出的问题，至今仍未解决。如果是这样，那么至少应该存在两个最小度数都很大的反完全子图，这就是我们的结果之一。首先是强化：我们可以要求两个子图中的一个具有大色度数：即对于所有 t,c≥1，存在 d≥1，使得如果 G 的色度数至少为 d，并且不包含完整图 Kt 作为子图，那么存在反完整子图 A,B，其中 A 的最小度数至少为 c，B 的色度数至少为 c。其次，我们来看看如果用 G 具有足够大的最小度这一假设来代替 G 具有足够大的色度数这一假设，会出现什么情况。这一点，加上排除 Kt，还不足以保证两个反完全子图都具有很大的最小度；但是如果我们不排除 Kt，而是排除完整的双向图 Kt,t，就能做到这一点。更确切地说：对于所有 t,c≥1，存在 d≥1，使得如果 G 的最小度至少为 d，并且不包含完整双方形图 Kt,t 作为子图，那么存在两个最小度至少为 c 的反完全子图。

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On a problem of El-Zahar and Erdős

Two subgraphs $A, B$ of a graph G are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results.

We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have large chromatic number: that is, for all $t, c \geq 1$ there exists $d \geq 1$ such that if G has chromatic number at least d, and does not contain the complete graph $K_{t}$ as a subgraph, then there are anticomplete subgraphs $A, B$ , where A has minimum degree at least c and B has chromatic number at least c.

Second, we look at what happens if we replace the hypothesis that G has sufficiently large chromatic number with the hypothesis that G has sufficiently large minimum degree. This, together with excluding $K_{t}$ , is not enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding $K_{t}$ we exclude the complete bipartite graph $K_{t, t}$ . More exactly: for all $t, c \geq 1$ there exists $d \geq 1$ such that if G has minimum degree at least d, and does not contain the complete bipartite graph $K_{t, t}$ as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least c.

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.