Back in $2017$ or so$,$ the Conn College registrar approached us with the following problem. The number of faculty holding scheduled final exams each semester had grown to the point where the registrar's office was spending countless hours at the end of each semester trying to figure out a way to put each exam into a scheduled time slot so that no student was scheduled to take two different final exams at the same time. It turns out that they were wrestling with an NP$-$hard problem; an efficient algorithm for scheduling the final exams into the fewest possible time slots could be used to efficiently solve the known NP$-$hard graph coloring problem, which asks: Given an undirected graph $G=(V,E),$ what is the minimum number of colors needed to color the nodes such that no two adjacent nodes have the same color? Please answer the questions below about the following reduction algorithm from the Graph Coloring Problem to the Exam Scheduling Problem.$($a$)(3$ points$)$ If the Exam Scheduling Black Box returns an exam schedule that uses only k time slots, what does that mean about the colorability of the graph G $?($b$)(3$ points$)$ If two nodes v$\_($a$)$ and v$\_($b$)$ in a valid graph coloring of G share the same color, what would that imply about final exams a and b $?($c$)(4$ points$)$ Consider the decision $($yes$/$no$)$ version of the above Exam Scheduling problem: Given a set of courses, with a roster of students for each course that needs to take a scheduled final exam for that course, and a number k$,$ is there a way to schedule exams into at most k time slots so that no student has to take two exams at the same time?For which value$($s$)$ of k is this an NP$-$hard problem? Check any that apply.$23$quad$4$quad$5$ and up