I haven't heard of "many" books on low carb plans allowing one day off a week. As a matter of fact, the only one I can think of is CKD where you have a "carb up" day and then follow it with very low carb and strenuous weight training the rest of the week. The purpose of it isn't just to allow those that follow it a pig-out day, but a calculated input of glucose to re-supply the muscles with glycogen for the purpose of increasing muscle mass. There are specific rules that go with it (high sugar/low fat) as well as the required workouts so if you're not planning to weight train strenously, it probably isn't for you.
Dr. Atkins did not encourage "cheat days" or "days off" since they will likely throw you out of ketosis and most people tend to take 2-3 days to get back into ketosis again leaving you only 3 days a week in which you are actually burning fat.