I heard the chants of phoenix sucks at the kings game, not sure if they were due to any rivalry or (more likely) just stating the facts at this point in time.
I think Kings fans do that to every team. The biggest rivalry for the Kings is the Sharks and Ducks. The Sharks/Kings rivalry will get some more fuel on the fire with them (likely) meeting up in the first round. I agree with everyone who said the Coyotes need to meet up with the Kings a few more times in the playoffs before there is a rivalry.