关于最小砖块的诺林-托马斯猜想

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-09-10 DOI:10.1002/jgt.23175

Xing Feng

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引用次数: 0

摘要

如果对每一对顶点都有一个完美匹配，那么一个三连图就是一块砖。如果对每一条边都不再是一块砖，那么这块砖就是最小的。诺林和托马斯证明了每块最小图至少包含三个三度顶点，并提出了一个更强的猜想：存在这样的情况，即每块最小图至少有三个立方顶点。在本文中，我们证明了这个猜想对于所有平均度数不小于 23/5 的极小砖块都成立。作为其推论，我们证明了每个极小砖块的顶点都包含度数最多为四的顶点以上。

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On a Norine–Thomas conjecture concerning minimal bricks

A 3-connected graph $G$ is a brick if $G - S$ has a perfect matching, for each pair $S$ of vertices of $G$ . A brick $G$ is minimal if $G - e$ ceases to be a brick for every edge $e \in E (G)$ . Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists $α > 0$ such that every minimal brick $G$ has at least $α ∣ V (G) ∣$ cubic vertices. In this paper, we prove this conjecture holds for all minimal bricks of an average degree no less than 23/5. As its corollary, we show that each minimal brick on $n$ vertices contains more than $n / 5$ vertices of degree at most four.

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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 .