No, it's not. Ft. Worth is run by Republicans. San Diego, Jacksonville, El Paso, Oklahoma City, Fresno, Mesa, Omaha, Colorado Springs, Miami, Virginia Beach, Tulsa, Arlington...all have Republican mayors.
Well in that case, the issue gets even more complex as you factor in whether or not a city with 4 Democrats and 3 Republicans on the council and 1 Republican mayor is a "Republican" city or a "Democrat" city.