One of the requirements for filing as head of household is that you paid more than half the cost of keeping up the home. The two adults can't both have paid more than half the cost. If one paid more than half, the other must have paid less than half. So only the one who paid more than half can file as head of household. The other one has to file as single.
People in this situation sometimes say that they split the expenses equally. If they truly split everything exactly 50-50, then neither can file as head of household because neither one paid more than half the expenses.
