Estate agent George Low, 22, from Dartford, was stabbed in the neck near a nightclub in Ayia Napa on Sunday.
Cypriot police said an argument broke out after the alleged attackers took exception to two British men urinating in the street.
A shoving match is then said to have broken out before Mr Low and his friend, Ben Barker, 22, were stabbed.
Mr Barker had knife wounds to his back in the attack in Grigori Axfentiou Street, but survived.
Arrest warrants have been issued for two Turkish Cypriot men, Mehmet Akpinar, 22, and 42-year-old Sali Ahmet.
Police have also issued a warrant for a third man suspected to have helped the suspects escape to the Turkish-held north of the holiday island.
A 48-year-old Greek Cypriot woman has been remanded in custody on suspicion of being an accessory to murder.
George Economou, deputy police chief of the Famagusta region, said officers had been told the two British men were urinating in the street and two Turkish Cypriot men told them not to do it in public, which led to an argument breaking out.
He said 11 minutes later, the men allegedly returned with a knife.
The group did not know each other beforehand, Mr Economou added.
He said police were continuing to look for CCTV to try to trace the route the suspects took to escape.
Checkpoints and airports have been put on alert, he added.
The woman who has been taken into custody pending further inquiries - but has not been charged - is said to be the girlfriend of Mr Akpinar and is accused of taking him a fresh pair of clothes and retrieving his mobile phone.
A district court in Larnaca remanded her in custody for eight days.
Mr Low's mother, Helen, 47, has said her son, a former club manager at the resort, was not an aggressive person and had simply been "in the wrong place at the wrong time".
She said he and Mr Barker had been due to fly home later the same day.
A British holidaymaker stabbed to death in Cyprus was involved in a row about urinating in public, police have said.