If you are living together unmarried, then only one parent can claim the child as a dependent, and that parent gets all the benefits. The other parent should not even list the child on their tax return.
If you are living apart (divorced or separated) and sharing custody, then the only person who can claim head of household status and claim the day care credit is the parent where the child lived more than half the nights of the year, regardless of any agreement. You can agree to switch the child tax credit around, but EIC, HOH and the dependent care credit always stay with the parent who had custody more than half the year, and can't be waived, transferred or shared.
