Talk about hot for teacher. Vivian is Grant's main squeeze. She's married but separated, trying to get a divorce, and has a couple of kids that she leaves with a babysitter in order to meet up with her boyfriend.
She is also a little bit of a scandal, because of the guys she finds herself attracted to. Vivian is not as dark-skinned as Grant and her first husband (at one point, Jefferson calls her "yellow"). And the community she is from takes that to mean that they are better than darker people. But Vivian rebels, first marrying a dark-skinned boy and then taking up with Grant.
This separates her from her own family, who quit speaking to her, and also from her potential in-laws, who see her as an uppity girl who thinks she's better than them because of her lighter skin.
Just like Grant, Vivian is a schoolteacher, but she works in Bayonne at a nicer school with more than one teacher and classrooms that separate the different levels of students. It's for black students only, because of segregation laws, but it is a lot nicer than the plantation school Grant teaches in.
Vivian acts a little bit like Grant's conscience within this novel. When Grant starts losing faith and talking gloom and doom, she won't let it slide and insists that he is committed to his students. For example, in their first meeting of the novel Grant is a little bit drunk and asks her if she wants to leave that very night:
"You want to go home and pack your clothes and get the children and leave from here tonight?"
She looked at me as though she was trying to figure out whether I was serious or not.
"No," she said.
[. . .]
"People do it all the time. Just pack up and leave."
"Some people can, but we can't," she said. "We're teachers, and we have a commitment." (4.65-71)
Grant's offer is crazy, but probably tempting given the difficult circumstances that they both face by staying in the Jim Crow South.
However, Vivian stands firm and refuses to accompany Grant. And since he doesn't really try too hard to change her mind, we get the idea that he needs her to refuse him and make him stay—that's her role in their little routine.
Vivian is the only one Grant doesn't really get mad at and he listens to her advice. She recognizes that he is committed to his community, just like her, even if he isn't willing to admit it.