Constructive modification of the Markov chain proof for balancing color sequences

Develop a constructive modification of the Markov chain–based argument showing that a uniformly random infinite color sequence on an Eulerian arc-colored directed multigraph balances arc visits for all pebbles almost surely, so that the method outputs an explicit infinite color sequence with the balancing property.

Background

In the pebble-on-graph abstraction, the authors prove that choosing colors independently and uniformly at random makes each pebble visit each arc with the same frequency almost surely (Lemma 4.1), but this is a non-constructive existence argument.

They explicitly note that the proof does not yield an explicit sequence and that it is unclear whether the probabilistic proof can be altered to provide one. Although an alternative constructive approach exists via Proposition 2 (periodic sequences), it does not arise from the Markov chain method and does not establish that almost all sequences have the property.