Q. We live in same residence. I have one child and my girlfriend has two children. we both split expenses equally. can we both claim HOH?
A. Simple answer: No. You are a family and a family can only have one Head of Household.
Room mate HoH
If you and your roommate can show that you each are maintaining separate households within the same structure. If you paid over one-half of the expenses attributable to yourself and your dependent children and She also paid over one-half the expenses attributable to herself and her dependent children. Even if you split the main costs like rent and utilities 50/50, but do separate food bills, and separate phone lines, you probably meet the requirement for "maintaining separate households".
Theoretically yes; but from a practical application, no. The rule is that you must have paid more than half the household expenses. If you paid half, the other person must have paid less than half.
If your situation closely matches the IRS "Acknowledged Significant Advice" at the link below, you probably qualify. http://www.unclefed.com/ForTaxProfs/irs-wd/1998/1998-041.pdf
It's a tough standard to meet and most living arrangements will not qualify. If it's family and not unrelated people, like 2 single moms sharing a residence, it's even more unlikely. If you try it, be prepared to prove it with detailed accounting records of household expenses.
https://www.thebalance.com/two-heads-of-household-3193038
