Palestinian president’s faction says dozens of members arrested in the Gaza Strip.
Both sides blamed each other for provoking the latest spate of violence and reports were conflicting as to which sides those killed supported.
Fatah said dozens of its activists were arrested overnight and the home of Ahmed Halas, the group’s senior leader in Gaza, was fired upon.
It also said Hamas security forces removed Fatah flags displayed on homes and detained Ibrahim Abu Najah, a Palestinian leader not affiliated with either of faction, shaving half his hair and moustache off as a form of humiliation.
Islam Shahwan, a spokesman for Hamas security forces, confirmed only that his men made a number of arrests.
However, a senior Hamas official admitted to Al Jazeera that there had been some excesses by security forces but insisted they were not condoned by the group.
The official, under the condition of anonymity, said young fighters were reacting to provocative statements made by Halas and Najah in a tense atmosphere further exacerbated by a crackdown on Hamas officials in the Fatah-run West Bank.
Kayed al-Ghul, a Popular Front for the Liberation of Palestine leader and mediator
Kayed al-Ghul, a PFLP leader helping mediate the conflict, said Hamas needed to keep better control of its security forces and called for an end to the Hamas crackdown on anniversary celebrations.
“We want Fatah’s anniversary to be a national celebration, not a factional celebration,” he said.
PLO factions mark the anniversary of Fatah as the start of the Palestinian national liberation movement.
Against this backdrop of factional tensions, Abbas urged Hamas to agree to early elections and to cede control of the Gaza Strip.
In the speech marking the Fatah anniversary on Monday, he said: “I renew the option of early elections … and I pledge that I will do my best to ensure this election will be the product of a deep and brotherly understanding.
“I urge all, Fatah and Hamas movements and all other Palestinian factions, to study this alternative and not to rush, as usual, to reject it.”
Hamas, which won a Palestinian parliamentary vote in 2006, opposes holding elections before they are due in 2010, citing that any polls prior to this date would be unconstitutional.
Meanwhile, at least one Hamas fighter was killed and several others wounded in an Israeli artillery bombardment of a refugee camp in the central Gaza Strip on Tuesday.
Hamas said the dead man was a member of its security force and the Israeli military confirmed the strike.
A Qassam rocket fired from Gaza fell in the Israeli town of Sderot but caused no damage.