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The divorce court must award community property equally between the parties. If the husband were given the house, the court must divide all other assets and debts so that the parties share the assets and debts equally.
Community property is nearly all property acquired during the marriage, other than gifts or inheritances.
Also, if the house were purchased with one party's separate property funds, the house could be a separate property asset.
It is common to give the custodial parent possession of the house (which is not the same as ownership of the house) while the children are under 18. That usually only happens if the custodial parent will be able to pay the mortgage (even if they need child support and alimony to do so.)
Assuming the house was acquired during the marriage, both parties have an equal interest in it. If the mortgage was acquired during the marriage, both parties owe it under community property rules, although if it goes unpaid, the mortgage holder would not report the default on the wife's credit report.
So he absolutely does not have an exclusive right to the house and the court will have to decide who gets possession and how the property and liability interests will be divided.
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