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Any two reduced expressions for the same Coxeter group element are related bya sequence of commutation and braid moves. We say that two reduced expressionsare braid equivalent if they are related via a sequence of braid moves, and thecorresponding equivalence classes are called braid classes. Each braid classcan be encoded in terms of a braid graph in a natural way. In this paper, westudy the structure of braid graphs in simply-laced Coxeter systems. We provethat every reduced expression has a unique factorization as a product ofso-called links, which in turn induces a decomposition of the braid graph intoa box product of the braid graphs for each link factor. When the Coxeter graphhas no three-cycles, we use the decomposition to prove that braid graphs arepartial cubes, i.e., can be isometrically embedded into a hypercube. For aspecial class of links, called Fibonacci links, we prove that the correspondingbraid graphs are Fibonacci cubes.

Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes