RV running? I assume it's a motor hope then? Having the engine running doesn't have that much to do with the refrigerator.
If you have a compressor style refrigerator it always runs on electricity. This can come from being plugged in, from the engine alternator, or from your battery.
If you have an absorption refrigerator then it always runs on heat. (I know, that doesn't make sense but it works.) The heat can come from burning propane, from an electric element powered by electricity from being plugged in, or from an electric element powered by your battery. The last choice is really not an option unless you have a very large battery and a very good charging system. An absorption refrigerator running on a battery uses A LOT of power. Your best option is to run it on propane when going down the road and on "shore power", electricity from being plugged in, when you are camped.
There are a few folks who have some sort of irrational fear of using the refrigerator on propane when they are driving. Don't worry about them. Probably 99% of RVers run it that way because that's the way the manufacturer designed it to be used.
If you have a new RV you likely have a compressor refrigerator but it's hard to say since you didn't give any details.