A convention of mathematicians will have rooms available in $6$ hotels. There are $n$ mathematicians and, because of personality conflicts, various pairs of mathematicians must be lodged in different hotels. The organizers wonder whether $6$ hotels will suffice to separate all conflicts. Model this conflict problem with a graph and restate the problems in terms of vertex coloring.
I am immediately confused about this question simply because I'm wondering are my vertices the hotels, the $n$ mathematicians, or the mathematicians with conflicting personalities? How could I model this using Graph Coloring?