No. When two unmarried parents live together, only one person may claim the child as a dependent. The special rules for "splitting" a child that allow one parent to claim the tax credit and the other parent to claim HOH and EIC only apply to parents who live apart and share custody.
If the parent who pays more than half the household expenses (usually the person with higher income but not always, it depends on who pays for what) claims the child, they get all the tax benefits including HOH and the other parent files single with no dependents. If the parent who pays less than half the household expenses claims the child, then both parents file as single. If the lesser-earning parent qualifies for EIC, that might offset the benefit to the higher-earning parent of filing as HOH, but the only way to know is to test both ways.
