text: Colonel Christopher Garver said Iraqi forces had cleared only a third of the city and the rest remains contested.
  Iraqi commanders insist that 80% of Falluja is under their control.
  Iraq's prime minister announced Falluja's "liberation" on Friday, after the city council building was retaken.
  Haider al-Abadi claimed that remaining pockets of IS fighters would be "cleaned out within hours".
  But clashes between government forces and IS militants continue and the militants have dug in in residential neighbourhoods in the north of the city.
  Falluja: Why the embattled city matters
  Devastated Falluja remains a battleground
  Fight to drive IS from Iraq far from over
  Islamic State group: The full story
  Iraqi forces moved into the centre of the city on Friday, raising the national flag from the city council building.
  But Colonel Garver warned that the troops would encounter more resistance as they moved out from the centre of the city.
  He said: "What it looks like is (an IS) defensive belt around the city with not as stiff defences inside. That could be their toughest fighting."
  Iraqi commanders said on Tuesday they had pushed IS out of two northern neighbourhoods of the city.
  They say the battle for Falluja has killed 2,500 militants, but the figure has not been independently backed up.
  The operation to retake the city has pushed thousands of residents to flee, overwhelming refugee camps.
  More than 83,000 people have fled since the government launched the assault and up to 25,000 more are likely to be on the move, according to the UN.
  The Norwegian Refugee Council said reception camps for the displaced were overwhelmed, and there was insufficient water, food, shelter and medical care.
  Falluja, only 50km (30 miles) west of the capital Baghdad, has been held by IS for longer than any other city in Iraq or Syria.
  The jihadist group overran the predominantly Sunni Arab city in January 2014, six months before it seized control of large parts of northern and western Iraq.
target: A US military commander in Baghdad has openly contradicted the Iraqi army's claim last week that it had liberated the key city of Falluja and driven out IS militants from most of the city.