Ohio Rev. Code § 5521.05
If the board of county commissioners or municipal authority, after adopting the maps, plans, profiles, specifications, and estimates, are still of the opinion that the work should be constructed, and that they should co-operate upon the basis set forth in the proposal, such board or municipal authority shall adopt a resolution requesting the director of transportation to proceed with the work, and shall enter into a contract with the state providing for the payment by such county or municipal corporation of the agreed proportion of the cost and expense. The form of such contract shall be prescribed by the attorney general, and all such contracts shall be submitted to the attorney general and approved by him before the director may advertise for bids. All improvements upon which any county or municipal corporation may co-operate shall be constructed under the sole supervision of the director. The proportion of the cost and expense payable by the county or municipal corporation, shall be paid by the county treasurer or the treasurer of the municipal corporation upon the warrant of the county or municipal auditor, issued upon the requisition of the director. Upon completion of the improvement the director shall ascertain the exact cost and expense thereof and notify the board or municipal authority. Any balance, in the fund provided by such board or municipal authority, shall be returned to the county or municipal corporation.
R.C. §5521.05