I am using the old flag in my office and at home," said the mayor of Fallujah, Saad Rasheed, adding he would raise the new one only if the Anbar provincial council told him to.
The US-backed al-Hurra television station on Saturday said the Anbar provincial council would not fly the new flag, but Khamis Ahmed, a senior member of the body, denied the report.
The phrase Allahu Akbar (God is Greatest), added in green Arabic script on Saddam's orders during the 1991 Gulf War, remains, but no longer in his handwriting.
Thousands of Iraqis lost their lives so this flag could fly ...
Changing the flag ignores their sacrifice," said one Iraqi in a comment posted on an Arab chat room. It is still red, white and black, but three green stars in the centre representing unity, freedom and socialism, the motto of Saddam's now outlawed Baath party, have been removed.
We refuse to change the flag because it represents us all," said Amir Saadoun, a resident of Baghdad.
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The new flag will last for only one year, while debate will continue on what the final banner should look like.