I would have your child clean up the graffiti but I hope the discipline does not stop there. She needs to understand that what she did was very wrong, frankly, and just cleaning up chalk isn't going to make that impression. I'd have her lose some privilege she truly values--you know best what that would be--and I'd have her lose it for a substantial enough time that she feels the loss and the seriousness of what she did. I would also have her write a letter of apology to the family (simple and short) so that she and this girl do not have to come face to face, but she does learn how to make a sincere apology. I'm not saying she's a bad kid, so please don't get that impression! I'm saying that she must take full responsibility here -- the other girl is a problem for sure, and does stuff she shouldn't, but your daughter did something that's utterly wrong -- and if your child says "she's as bad, she pushed me to do it by being mean to me," that is no excuse for her own behavior. This is a huge learning opportunity for your child.
Then continue with your very wise tactic of having your child utterly ignore this other girl. You mention that the other girl yells rude things when your kids get off the bus -- does this girl go to their school or not? If she does, I would bet that this battle goes on there as well. I'd be glad it's summer and I'd also ensure that this girl is not in any classes with your kids next year, if that's a possibility at all. Many schools let parents write letters to the main office identifying kids who should or should not be placed with their own children. The schools don't have to honor that, but some truly do (ours would, if you explained that there are serious issues at home between the kids).
As for the mom, once the letter of apology goes over there, I would just keep avoiding her. If she persists in talking to you after your child is disciplined and has apologized, and if she keeps on harping on you when you see her - then she's an adult bully and you may need to simply tell her, "My child has been seriously disciplined and told to ignore your child so that they do not 'get into it' with each other. It's best if they have a very real break and don't talk until they both are more mature. We'll go our way and you go yours and maybe in several years the girls can be better neighbors but we will give them a big breathing space." I would hope she'd understand that "go your way, we'll go ours" means the adults, too.