Two adults cannot claim head of household for maintaining the same household since two adults cannot pay more than half the cost of keeping up the same home.
To claim HOH status-
To qualify for head of household status, you must be either unmarried or considered unmarried on the last day of the year. You are considered unmarried on the last day of the tax year if you meet all the following tests.
You file a separate return
You paid more than half the cost of keeping up your home for the tax year.
Your spouse did not live in your home during the last 6 months of the tax year. Your spouse is considered to live in your home even if he or she is temporarily absent due to special circumstances.
Your home was the main home of your child, stepchild, or foster child for more than half the year.
You must be able to claim an exemption for the child. However, you meet this test if you cannot claim the exemption only because the noncustodial parent can claim the child using the rules