For the next Olympic Winter Games the organizers wish to exp

For the next Olympic Winter Games, the organizers wish to expand the number of teams competing in curling. They wish to have 14 teams enter, divided into two pools of seven teams each. Right now, they’re thinking of requiring that in preliminary play each team will play seven games against distinct opponents. Five of the opponents will come from their own pool and two of the opponents will come from the other pool. They’re having trouble setting up such a schedule, so they’ve come to you. By using an appropriate graph-theoretic model,either argue that they can not use their current plan or devise away for them to do so.

Solution

Use the Degree Sum Formula, which states that for all graphs

G(V,E), vVdeg(v)=2|E|

This implies that \"For any graph, the number of vertices with odd degree is even.\" (Corollary 5.2 in the book).

In a specified pool, each of the 7 teams must play exactly 5 others in the same pool. As a graph, each of the teams is a vertex with degree of exactly 5. This is impossible, as the number of vertices with odd degree is odd.