In a university, the secretariat plans the examination period. There are $6$ subjects, $A,B,C,D,E,Z$ and $9$ students($1, \dots , 9$). At the subject $A$ the students $1,2,3$ are subscribed, at the subject $B$ the students $1,2,9$, at the subject $C$ the students $1,7,8$, at the subject $D$ the students $3,5,7,9$, at the subject $E$ the students $4,5,8$ and at the subject $Z$ the students $4,6,8$. Each examination lasts $2$ hours, and it can only be during the morning hours $10-12$. The only restriction at the planning is that it is not allowed that $2$ subjects, ,at which the same student is subscribed , get examinated simultaneously.
Which is the minimum number of days that are required,so that all the exams are taken?
I tried to solve the exerise,with the chromatic number,using the following graph:
So,the minimum number of days is $4$..or am I wrong??
I added labels to the graph's edges, for anyone having trouble seeing what's going on.
In addition, here is a table of the classes and students. $$ \newcommand{\x}{\blacksquare} \begin{array}{|c|cccccc|} \hline & A & B & C & D & E & Z \\ \hline 1 & \x & \x & \x & & & \\ 2 & \x & \x & & & & \\ 3 & \x & & & \x & & \\ 4 & & & & & \x & \x \\ 5 & & & & \x & \x & \\ 6 & & & & & & \x \\ 7 & & & \x & \x & & \\ 8 & & & \x & & \x & \x \\ 9 & & \x & & \x & & \\ \hline \end{array} $$