Why didn't the Thebans chase after Laius's murderer?
The Thebans probably did not chase Laius' murderer because they were distracted by the lifting of the plague laid on the city of Thebes by the Sphinx. When Oedipus arrived in the city and solved the riddle of the Sphinx (What goes on four legs in the morning, two legs at noon, and three legs in the afternoon? Answer: man), the plague that the city has suffered is gone, and the city rejoices. They reward Oedipus by giving him the queen Jocasta in marriage and making him their new king.
Another reason may be that the only man who survived the attack at the place where three roads meet when Laius was killed reported that a band of murderers killed the king and his entourage. Only one man escaped. This man chooses to leave Thebes to become a herdsman in the mountains without telling the truth, that only one man--the man who has just been crowned king--was responsible for Laius' death. The citizens of Thebes may figure that attempting to locate all of this so-called band of murderers is a futile search.
More than likely, however, the reason is their celebration because the deadly plague is over.