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arXiv · 2405.19543

Coloring minimal Cayley graphs

Ignacio García-Marco·Kolja Knauer
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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.

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Ignacio García-Marco, Kolja Knauer. 2024-05-29. Coloring minimal Cayley graphs. https://arxiv.org/abs/2405.19543

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