The official line from Cambridge Diet is..."It is best not to because chewing gum stimulates the release of gastric juices which in turn will make you feel hungry, if you must chew, chew for 2 or 3 minutes if you feel you have to.
Also, even in the sugarless gum you have 5 calories a stick.
Always worth trying something like Goldspot breath spray, or even the little strips that melt on your tongue. But as with anything you add to a VLCD you need to be careful that it (a) doesn't kick you out of ketosis, or (b) start off cravings!
Perhaps a travel toothbrush and mini toothpaste for a quick brush up in the ladies?