For a child over 18, the special rules for divorced or separated parents no longer apply, and the dependent status is based on the normal rules without regard to the divorce.
To claim a child as a dependent, there are two methods.
A. the child lives with you more than half the year and does not pay more than half their own expenses.
B. the child has taxable income less than $5050 and you pay more than half their expenses.
However, only the first type qualifies for HOH, they are not a qualifying person for HOH unless they live with you more than half the year. Only one parent can use the child for HOH. It can't be split or shared. For a child at college, they are usually considered to "live at home" and college is considered a temporary absence. "Home" would be wherever the child spends the majority of vacations and breaks, where they keep their stuff they haven't taken to school, and where the child is most reasonably expected to return if they "come home" after graduation while they figure out their next step.
I see two problems here. If the child has a year-round apartment, they might not live "at home" with either parent, in which case neither parent can claim HOH status. (This is not a black letter rule. If you paid for an apartment instead of a dorm, but the child is planning to come home after graduation, then they might still be considered to "live at home" and be temporarily at school. It depends on the facts unique to each case.)
Then, even if the child can arguably be considered to live "at home" more than half the year, that can only be with one parent. If you think that is you, you can file as HOH (but you may need to mail your return) and then the IRS will investigate both parents and make a determination based on evidence as to where the child really lives and if anyone can claim them as a dependent. The losing parent (or parents) will have to recalculate their tax return and pay back any excess refund, with interest.
