Coloring minimal Cayley graphs

IF 1 3区 数学 Q1 MATHEMATICS
Ignacio García-Marco , Kolja Knauer
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引用次数: 0

Abstract

In 1978 Babai raised the question whether all minimal Cayley graphs have bounded chromatic number; in 1994 he conjectured a negative answer. In this paper we show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 colors are sometimes necessary for soluble groups. On the other hand we address a related question proposed by Babai in 1978 by constructing graphs of unbounded chromatic number that admit a proper edge coloring such that each cycle has some color at least twice. The latter can be viewed as a step towards confirming Babai’s 1994 conjecture – a problem that remains open.
最小Cayley图的着色
1978年Babai提出了是否所有极小Cayley图都有有界色数的问题;1994年,他推测出一个否定的答案。本文证明了(有限生成)广义二面体群或幂零群的极小Cayley图的色数最多为3,而可溶群有时需要4色。另一方面,我们通过构造无界色数图来解决Babai在1978年提出的一个相关问题,这些图承认有适当的边着色,使得每个循环至少有两次颜色。后者可被视为朝着证实巴贝1994年的猜想迈出的一步——这个问题仍未解决。
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来源期刊
CiteScore
2.10
自引率
10.00%
发文量
124
审稿时长
4-8 weeks
期刊介绍: The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.
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