On cuts of small chromatic number in sparse graphs

Key Points

• A significant finding shows that the conjecture about chromatic number and separators fails dramatically.
• The study establishes that the lower bound for chromatic numbers is tight as $ frac{k}{2}$ for large graphs.
• Key evidence indicates that as $k$ increases, $ heta(k) o (1 + eta) frac{k}{2}$ holds asymptotically.
• The implications warn of complexities for identifying separators in large sparse graphs.