For either of you to claim Head of Household, one of the children must live with each of you more than half the nights of the year so yes you would need a custody agreement to ensure that the person who wants to file as Head of Household has one of the children living with him or her more than half the nights of the year.
Here is how it works:
For divorced or separated parents or parents who live apart, the custodial parent, if eligible, or other eligible person who the child lived with for more than half the year (183 nights in 2019), can claim head of household filing status, the credit for child and dependent care expenses, the exclusion for dependent care benefits, and the earned income credit. The non-custodial parent, if allowed by divorce decree or consent of the custodial parent on form 8332 or similar signed statement, can claim the dependency and child tax credit. For post-2008 divorce decrees or agreements, form 8332 or similar signed statement is required. The child tax benefits cannot be split any other way.
Here are all the requirements to file as Head of Household:
1. You are not married or “considered unmarried” on the last day of the year. You are considered unmarried if you lived apart from your spouse the entire last 6 months of the year.
2. If "considered unmarried" you file a separate return from your spouse.
3. You paid more than half the cost of keeping up your home for the year.
4. You have a qualifying child (whether or not you claim the dependency exemption) or you claim a dependency exemption for a “qualifying person” who is related to you and who lived with you in the home for more than half the year. An exception is a parent does not have to live with you to be a qualifying person.
