What if I don't buy airplane tickets for my girlfriend?
That depends. Why don't you want to buy her tickets?
I like her but she should buy airline ticket by herself
While people should buy their own things, it is good to help your partner if they can't afford it. Otherwise it can cause problems in the relationship.
I don't want to make her feel like she owes me.
That is good, but sometimes relationship do require for a partner to help one another out. Otherwise she might reconsider the relationship if she might as well be alone.