Are the rules for counties in local government spelled out in the US Constitution

Well yes, and no, the US Constitution dosent say much about local government however, the 10th amendment does cover the fact that local governments have reserved powers not specifically taken by the federal government.

Explanation:

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the virginia plan called for three branches of government and two houses of congress. representation in each house would be determined by population. the new jersey plan called for three branches of government and a single house of congress. each state would have an equal vote.