4.55 A country has established such an air network: any city is connected by air routes to at most 3 other cities, and from any city to any other city, at most one transfer is required. How many cities can this country have at most?
[Solution] From a city $A$, one can reach at most 3 cities, and each of these 3 cities can be connected to at most 2 other cities. It is evident that there can be at most 10 cities nationwide.

The graph on the right shows 10 cities and 15 flight routes, where each city is connected to exactly 3 other cities and meets the requirement of at most one transfer. Therefore, the maximum number of cities is 10.