1-2猜想适用于正则图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B Pub Date : 2025-05-19 DOI:10.1016/j.jctb.2025.05.002

Kecai Deng , Hongyuan Qiu

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引用次数: 0

摘要

1-2猜想断言每个图的顶点和边都可以在{1,2}中赋予权重，使得相邻的顶点得到不同的加权度。虽然这个猜想在一般情况下仍然是开放的，但已经证明可以使用权值集{1,2,3}来实现这一点。我们证明了权值集{0,1}对每个图都是足够的。作为一个推论，对正则图证实了1-2猜想。此外，对于正则图，我们验证了另一个关于局部不规则全着色的相关猜想。

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The 1-2 conjecture holds for regular graphs

The 1-2 conjecture asserts that the vertices and edges of every graph can be assigned with weights in

{1, 2}

such that adjacent vertices receive distinct weighted degrees. While this conjecture remains open in general, it has been proven that it is possible to achieve this using the weight set

{1, 2, 3}

. We demonstrate that the weight set

{0, 1}

suffices for every graph. As a corollary, the 1-2 conjecture is confirmed for regular graphs. Additionally, we verify another related conjecture concerning locally irregular total colouring, for regular graphs.

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来源期刊

Journal of Combinatorial Theory Series B 数学-数学

CiteScore

2.70

自引率

14.30%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.