What is the ruling regarding a husband who swore to divorce his wife if she left the house, and then she left and returned after a week?

If you swore to your wife that if she leaves the house she will not return to it, then if you intended to effect a divorce, the divorce occurs according to the majority of scholars and those who agree with them due to the presence of intention. However, if you did not intend divorce, then nothing is binding upon you.

If you swore by one of the names of Allah Almighty or one of His attributes, or you said: "I swear by Allah Almighty," and you broke your oath by your wife returning, then an expiation for breaking an oath (kaffarat yamin (definition)) is binding upon you. But if you only said: "I swear," your oath is not considered valid according to the majority of scholars, and there is no expiation upon you.

Divorce and breaking an oath by Allah do not combine upon you. So, if you intended divorce, it has occurred, and you are not considered to have broken an oath. If you did not intend divorce, then you are considered to have broken an oath by Allah, but the wife is not divorced.

If a divorce occurs, you can take your wife back before her 'iddah (definition) (waiting period) ends if it was the first or second divorce. Her 'iddah ends upon her purification from the third menstrual cycle after the divorce, or after the passage of three months, or upon giving birth if she is pregnant. If her 'iddah has expired, then a new marriage contract is necessary. However, if this was the third divorce, then she is unlawful for you until she marries another man and he divorces her after consummation.