Q:

Scheduling and coloring problem

A group of students plan to take part in exams in the courses ${C_1}, \ldots ,{C_6}$ as follows:
$$\begin{array}{l}
{\rm{Student A}} \hspace{7 mm} {C_1}\,\,\,\,\,{C_3}\\
{\rm{Student B}} \hspace{7 mm} {C_1}\,\,\,\,\,{C_4}\,\,\,\,\,{C_5}\\
{\rm{Student C}} \hspace{7 mm} {C_2}\,\,\,\,\,{C_6}\\
{\rm{Student D}} \hspace{6.8 mm} {C_2}\,\,\,\,\,{C_3}\,\,\,\,\,{C_6}\\
{\rm{Student E}} \hspace{7.2 mm} {C_3}\,\,\,\,\,{C_4}\\
{\rm{Student F}} \hspace{7.2 mm} {C_3}\,\,\,\,\,{C_5}\\
{\rm{Student G}} \hspace{6.8 mm} {C_5}\,\,\,\,\,{C_6}
\end{array}$$
Draw a graph with nodes ${C_j},\,j = 1,2, \ldots ,6$ so that there is an edge between $C_j$ and $C_k$ if and only if there is at least one student that plans to take part in the exams in courses $C_j$ and $C_k$. Determine the chromatic number of this graph, that is the smallest number of colors needed to color the nodes so that if there is an edge between two nodes, then they get different colors. What does this number tell us in this case?
My attempt:
One of the possible graphs:
         
I have already colored it, and i know how to explain the way i did it. From this graph we see that $\chi{(G)} = 3$ and to prove that the chromatic number cannot be smaller than $3$ we just analyse the topmost cycle, which requires minimum $3$ colors.
My question is regarding the graph. Did i get it right? Does it satisfy all conditions? My doubts are regarding the students that take $3$ exams. I hope i don't need to draw the edge, say for example for student $B$, $\left( {{C_4},{C_5}} \right)$, because from the edges $\left( {{C_1},{C_4}} \right),\,\left( {{C_1},{C_5}} \right)$ we already see that student $B$ will take exams ${C_1},\,{C_4}$ and ${C_5}$, but i'm not sure if i'm right?

A:

Since there is a student (student B) taking both exams $C_4$ and $C_5$, the definition you're given says that there should indeed be an edge $(C_4,C_5)$. Note that the coloring you provided puts these exams at the same time, which is bad news for Student B.

