The first attack took place around midnight (2100 GMT) in al-Hawija, west of the oil centre, when an explosion ripped apart an army patrol, police Colonel Ahmad al-Ubaidi said on Monday.
Two soldiers were killed and eight wounded in the overnight blast.
Early on Monday, two more Iraqi soldiers were killed and two wounded in similar circumstances as they patrolled a town south of Kirkuk.
The city is an oil hub that many Kurds want as the capital of their autonomous region in northern Iraq.
A car bomb exploded near an Iraqi checkpoint in western Baghdad on Monday, killing three policemen and three civilians, police said.
The attack, which occurred around 11am near Al-Nusur Square, also seriously wounded two nearby Iraqi construction workers, said police Captain Qasim Husain.
In the capital's northwestern al-Huriya neighbourhood, police discovered five bodies early on Monday in a park, an Interior Ministry source said. Their hands were bound and the victims had been shot and killed.
US patrols are the target of daily
attacks in Baghdad and beyond
A mortar round was lobbed into the heavily protected administrative Green Zone in Baghdad, the source added, but details on any casualties were not immediately available.
In Hilla, a mainly Shia town south of Baghdad, another mortar round landed "in the vicinity of a US regional embassy compound", at 2am, US spokesman Vicki Stein said.
No one was injured in the incident, she added.
In other developments, Mishaan al-Jiburi, a member of the Iraqi National Assembly, told Aljazeera the Arab League's intervention in the Iraqi political process had resulted in approving the go ahead of a conference of Iraq's national forces.
Al-Jiburi believes the conference could lead to a set of recommendations suggesting that parliament form a constitution assessment committee.
Under the new recommendations
a new committee may be formed
"A conference will be held at the headquarters of the national assembly to amend the draft constitution," al-Jiburi said.
"This conference will issue a codicil of the constitution draft, which will be considered as a temporary constitution.
"It gives the parliament, which will be elected at the end of this year, the right to form a committee that assesses the constitution and suggests alternatives or drafts a new constitution," he said.