A math professor is trying to offer an introductory math class to students. Consider the following problems.

A math professor is trying to offer an introductory math class to students. Consider the following problems (a) Alice won't take a math class with Bob, Bob won't take a math class with Charlie, Charlie won't take a math class with David, David will not take a math class with Edward and Edward will not take a math class with Alice. To make matters worse, Fiona will not take take a math class with David or Alice. At least how many times must the math professor offer this class to accommodate the aforementioned set of 6 students? (b) For an arbitrary set of students (not necessarily the same set as for part a), when would the math professor only have to offer his class at two separate times? (c) When would the aforementioned professor have to offer his class separately once for each student! (d) A new extremely accommodating school first asks students what classes they would like to take for the term. Then once the registrar has this list of desired classes, the registrar determines the minimum number of time slots necessary for the courses such that no students have any scheduling conflicts. If one student take two different courses, those two courses must be scheduled at different time slots, and each course will be offered only once. First, translate the registrar's problem into graph theoretic terms and then determine how the registrar can determine the minimum number of slots needed