"As a result of military actions in Donetsk on October 10, three civilians died and four more were injured and hospitalised," said the mayor's office in Donetsk, the largest eastern city.
All but completely controlled by the pro-Russian separatists, Donetsk has come under increased crossfire in recent days as rebels and the Ukrainian army battle over its airport.
Another victim was an elderly man killed by a shell that hit his house near the town of Popasne in the Lugansk region, the pro-Kiev governor of the region, Gennadiy Moskal, said on his website.
One border guard was killed about 20 kilometres (12 miles) south of Donetsk when a convoy of border guards was fired upon, military spokesman Andriy Lysenko said today.
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More than 300 people have died since a ceasefire was signed on September 5, according to the United Nations. The Ukrainian authorities report 120 dead civilians and Ukrainian troops, without including rebel casualties.
Under the truce, both sides must pull back heavy artillery by 15 kilometres, creating a 30-kilometre wide demilitarised zone. However, that has yet to happen, especially around Donetsk international airport, where Ukrainian units have held out against rebel attacks since May.
"We are ready to withdraw heavy weaponry as it was agreed in the Minsk deal," Lysenko told journalists. "We have prepared all territories and routes but we are waiting for the terrorist side to do this."
Donetsk rebels say they are ready for a withdrawal -- but again, only if the other side fulfills its side.