No. Byzantine is an adjective connoting the eastern part of the Roman Empire, from the city named Byzantium, its seat of government. Chariot races were held in a circus in the west and in a hippodrome in the east.
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In order to participate in a chariot race one had to be a member of a team. Rome had four teams, the Reds, Whites, Blues and Greens. It's unclear as to exactly how many entries each team could have in a race, or if the various teams raced against each other in a single race or in a matchup.
2012
In Ancient Rome the chariot races were team affairs. The Reds, Blues, Greens, and Whites would race against each other.
Chariot Races
An hippodrome.