Hi, it's me again - this time for real. The above is an automated message! 
Welcome to the site; I really hope you enjoy it!
Many people have asked very similar questions about bleeding here before, so please do search the forums before starting threads on similar topics. We try to avoid redundant topics whenever possible. The Advanced Search function under the magnifying glass icon at the upper right of this site is super powerful; I've copied a link below to what came up when I searched just the forums for the word "bleeding". Do check out some of these posts; I'm sure they'll be helpful.
In the meantime, you did bring up a new thought here, which I think merits separate discussion and that is the one of adding white food coloring (to prevent bleeding??). I've actually found that adding white food coloring delays the drying of icing (perhaps this has something to do with all of the titanium dioxide in it), in some cases quite a long time. So I suspect that contributed to the bleeding. The longer the icing takes to set, the greater the risk of bleeding. I simply work with icings as thick as possible for the task at hand (again, the thicker the icing to start, the faster it dries) and I rarely get much, if any, bleeding even with brown and black next to white icing (with no additional white coloring in it).
Hope this helps a bit. And, again, please do search for related posts next time, and check out all of these: