There IS a contract. But it has nothing to do whatsoever with and makes no mention of your work hours. It's called 'extending an offer of employment' which you accept or decline that states that they can terminate you at any time for a list of reasons, and that you are legally able to quit at any time for any reason.
In America generally speaking you will know beforehand whether or not you will be working full time or not, which is any work week that is 32+ hours of work. So obviously part time would be anything less than 32 hours per week.
At least, this is how it is in Oregon.... But I'd assume the laws in other states are similar
Edit- and as another prior user mentioned, a job that has a salary actually requires a certain amount of hours per week of the employee which would require a separate contract. These are jobs that are more often than not 40+ hours per week. It's a way to get around paying overtime legally, but salary jobs also pay more overall.
