When the weather is cold you should really leave both the hot and cold water lines dripping to keep the water flowing to both of them.
This is especially true if you have any hot water lines exposed underneath your home to the cold air.
Hot water will actually freeze faster so if you don't leave your hot water dripping through the night as well then you'll likely wind up with frozen hot water pipes as well.
When it's winter I leave just the cold water dripping while I'm awake or home but if I'm asleep then I will let the hot water drip along with the cold water to prevent it from freezing.