Is it obligatory to inform the mother about her daughter's communication with a young man via the internet and phone, even though the daughter made her friend swear not to?

You must strive to deter your friend from following the footsteps of Satan before it is too late. Advising one another among Muslims, cooperating in righteousness and piety, enjoining good, and forbidding evil are all incumbent duties. If you can convince her without revealing her secret, that is best. Otherwise, you can seek help from someone who influences her or speak with her mother about the importance of supervising her daughter and monitoring her phone and computer.