What is the vow?
The vow is more thoroughly explained in Rise of the New Masters.
Here is a brief section about it:
A kiss is a vow. Outside of one’s family, a person does not kiss another unless that person is vowed to the other. After the dragons died out at the end of the Dragon Era, sword masters and wizards alike had difficulty in controlling leaking of their magical powers. They learned to concentrate the distribution of their magic power through their lips—whether in words or in a kiss.The Age of Merrik