Ukraine's president vowed today to reassert government control over eastern regions as the army unleashed a counter-offensive against Russian-backed separatist fighters vying for command over the airport in the city of Donetsk.

The separatist stronghold was shaken by intense outgoing and incoming artillery fire over the weekend as a bitter battle rages for the air terminal and surrounding areas.

Streets in Donetsk, which was home to one million people before unrest erupted in spring, were completely deserted Sunday and the windows of apartments in the center were rattled by incessant rocket and mortar fire.

The warring sides exchanged rocket fire along several points in the roughly 220-mile front line.

Regional authorities loyal to the government said two children, aged seven and 16, were killed when a rebel shell hit their home in Vuhlehirsk, a town 45 miles north east of Donetsk.

A little further west, in the rebel-controlled city of Horlivka, two were killed and another 16 injured as a result of rocket attacks, city council secretary Oleg Gurbanov said in a statement.

President Petro Poroshenko told a crowd of several thousand gathered in the centre of the capital, Kiev, that Ukraine would not "give up an inch" of its land to Russian-backed separatists.

Hundreds of people, including Mr Poroshenko, held up signs reading Je Suis Volnovakha, referring to the 13 people killed when a passenger bus was shelled in the eastern town of that name. The message was an echo of the Je Suis Charlie tribute adopted after the terrorist attacks in Paris.

The separatists upped the ante last week by successfully taking over large sections of Donetsk airport, where Ukrainian troops remained despite coming under rocket attacks for months on end. Both sides have incurred losses in the close-quarter combat.

The rebels' progress has sparked a desperate Ukrainian fight-back supported by a hasty reinforcement of troops and heavy armoury.

Yuriy Biryukov, an adviser to Mr Poroshenko, said on his Facebook account today that Ukrainian troops had received orders to unleash heavy shelling of known rebel positions.

"Today we will show just how much we can smash their teeth in," Mr Biryukov wrote from a location near the fighting.

Separatist statements indicate Ukrainian forces may have attempted to burst into Donetsk itself, the first such effort since the unrest started in the spring.

The self-declared Donetsk People's Republic breakaway government said its forces had repelled a Ukrainian advance toward a bridge leading from the airport into the center.

Ukrainian military spokesman Andriy Lysenko said today that four servicemen were killed in the previous day's clashes. The office of the General Staff said separately that three troops had been killed in the airport alone.

A new truce agreed in early December unravelled one week into the new year despite concerted international efforts to forge a lasting settlement. High-level peace talks expected to take place in Kazakhstan last Thursday were postponed indefinitely.

The US has accused separatists of occupying territory beyond the line of contact agreed upon after a much-violated cease-fire deal in September. Russia, which has been unambiguous in its diplomatic support of the separatist stance, has accused the Ukrainians of weakening the prospects of that ceasefire deal reached in the Belarusian capital, Minsk.

In a sharply worded statement today, the Russian Foreign Ministry called for an urgent ceasefire.

The ministry said Russian president Vladimir Putin sent Mr Poroshenko a message last week about the necessity for both sides to respect the ceasefire and pull heavy artillery back from the line of contact.

Russia was "ready to use its influence on the separatist fighters to convince them to accept this version", the statement said.

Instead, the Ukrainian forces renewed their shelling of Donetsk, the statement said.

In the largest single loss of civilian lives so far this year, 13 people were killed after a bus parked at an army checkpoint near the town of Volnovakha was hit by a shell on Tuesday. Ukrainians swiftly accused separatists, who denied responsibility.

A fact-finding team from the Organisation for Security and Cooperation in Europe said on Friday evening that the shells had been fired from a "north-northeastern direction." The Ukrainians said that assessment confirmed their suspicions as the area in question is under rebel control. Russia's Foreign Ministry insisted the Osce report undermined Kiev's claims, but did not explain how.

PA Media