Ohio Rev. Code Ann. § 5557.06

The board of county commissioners shall receive bids and let the contract for improving such portion of a road as lies within the municipal corporation, in connection with the remainder of a proposed road improvement or separately, as the board determines. The total cost of such work shall be paid on the allowance of the board, by the warrant of the county auditor, and after completion of such work and the payment of the cost thereof, any balance of the funds contributed by the municipal corporation shall be refunded to it to be disposed of according to law.