The police chief in Paktia province, where the attack happened, said the bomber detonated his explosives around midday.
One month earlier, several members of an Afghan family died in an air strike in Paktia province.
American garrisons in places such as Gardez, the capital of Paktia province, already labour under rocket attacks.
Intriguingly, provinces where tribal structures are strongest, such as Paktia, Paktika and Khost, have proved most resistant to Taliban encroachment.
Chapman was killed on a mission to coordinate with local tribal elements in the Paktia province near the town of Khowst in eastern Afghanistan.
While the allied and Afghan forces pressed into Paktia's mountains, three Afghan fighters died Thursday in Kandahar, the former Taliban stronghold in southern Afghanistan.
In the Sayed Karam district of Paktia province in the east, coalition forces attacked Taliban militants Monday, said Esmatuallah Alizai, Paktia's provincial police chief.
The new interior minister, Taj Mohammed Wardak, the governor of Paktia, is an experienced former governor, though some questioned whether he will be tough enough.
Yusaf Khan, a Paktia University student who said he witnessed the Gardez airstrikes, said 28 Afghan civilians, including women and children, were wounded in the coalition attack.
Nato spokesman Lt Col Jimmie Cummings acknowledged that coalition forces had been conducting an operation in Paktia province "against a large number of insurgents" on Saturday night.
Loya Paktia remains startlingly egalitarian and determinedly suspicious of outsiders.
In May 2008, Sgt Petry's unit of elite Army rangers approached a house in Paktia province in eastern Afghanistan which intelligence suggested held high-value al-Qaeda officials, the US military said.
BBC: Afghanistan veteran Leroy Petry given US Medal of Honor
They believe it is by disrupting the Haqqanis' supply and infiltration routes in the eastern provinces of Ghazni, Paktia, Wardak and Logar that the battle for Kabul will be won.
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