Both Arizona State University and University of Arizona students celebrated their commencement on Monday with ceremonies in the Phoenix area. Thousands of students will graduate this week.

Arizona counties will continue struggling to fix roads, staff sheriff patrols and make payroll next year because the state has pinched their revenue since the Great Recession. But the Arizona Legislature’s budget for next year offers some relief.

Counties will fork over about $32 million to help balance the state budget that Gov. Doug Ducey was expected to sign this week.

The hit to counties includes payments they will be required to make to support state agencies such as the Departments of Revenue and Juvenile Corrections, and cuts to shared revenue such as the gas tax that counties previously received from the state.

The silver lining: The impact on counties of this year’s state cost shifts is about $20 million less than local officials feared.

“The counties are grateful for the relief that the Legislature and governor provided,” said Craig Sullivan, executive director of the County Supervisors Association of Arizona, a research and advocacy group representing the state’s 61 county supervisors.

“We go one year by one year, and see what we’re able to get accomplished. We’re generally pleased with the result.” The relief to counties includes: » $10 million in ongoing funding from gas taxes and license fees from the Highway User Revenue Fund.

» $8 million in one-time assistance for cost shifts related to the state Juvenile Corrections Department.

» $1.65 million in one-time Arizona Lottery funds shared with Mohave, Yavapai and Yuma counties.

Another win: Lawmakers agreed to fund construction at state universities without dipping into sales-tax revenue earmarked for counties and cities, as proposed originally.