Problem $1$

Problem $2v$

Problem $3v$

Problem $4\sqrt[\phantom{2}]{?}$

Problem $5$

Problem $6$

Problem $7$

Problem $8$

Problem $9...$

Problem $10$dots

Problem $11\sqrt[\phantom{2}]{?}$

Problem $12$dots

Problem $13\sqrt[\phantom{2}]{?}$

There are $10$ people enrolled in the Extended Learning classes. They are taking the classes as shown in the table. What is the minimum number of final exam times needed so that the people can take all of their final exams? Note that in the graph, the vertices are classes and the edges represent classes that are shared by someone.

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$\backslash$table$[[$Gambit$,$Business, Metalwork, History, Spanish$],[$Batman$,$History, Physics$],[$Daredevil$,$Astrology, Astronomy$],[$Hulk$,$History, PE$],[$Logan$,$Astrology, Metalwork$],[$Magneto$,$Metalwork, Mathematics, Physics, Latin$],[$Iceman$,$History, Metalwork$],[$Alfred$,$Spanish, History, Metalwork, Mathematics$],[$Elektra$,$PE$,$ Mathematics$],[$Klingon$,$Astrology, Physics, History$]]$

In the following table, indicate in which final exam time each class could be placed $($i$.$e$.1,2,$ etc$).$

$\backslash$table$[[1,$Metalwork$],[2,$Spanish$],[3,$Physics$],[1,$PE$]]$

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