Coloring minimal Cayley graphs

TL;DR

This paper investigates the chromatic number of minimal Cayley graphs, showing bounds for certain groups and constructing examples with unbounded chromatic number, advancing understanding of Babai's conjectures.

Contribution

It proves that minimal Cayley graphs of generalized dihedral and nilpotent groups have chromatic number at most 3, and constructs graphs with unbounded chromatic number with special edge colorings.

Findings

Minimal Cayley graphs of generalized dihedral groups have chromatic number ≤ 3

Some soluble groups require 4 colors for minimal Cayley graphs

Constructed graphs with unbounded chromatic number and special cycle colorings

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.