Who flies the planes themselves - do you contract with the same airlines who run regional flights for the national carriers? (What's the planes' livery?)

Part of why I'm curious is that I've noticed that the national carriers in the US have an extremely safe record of late - I believe there has been one death from 2002-present from the national carriers (Southwest 1248 slid off a snowy runway in Chicago, hit two cars, and killed one child), but there have been a many fatal accidents with regional or private carriers, so I'd like to know who the actual operating carrier is and what their safety record is.