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By Dieter Jungnickel (auth.)

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Back to tournaments again! Basically, any factorization of K 2n can be used. But in reality, in most cases there are additional conditions the schedule should satisfy. For example, the teams should play an equal number of times on each of the n playing fields (because they might vary in quality, for example). The best one can ask for in a tournament with 2n-l games for each team is, of course, that each team plays twice on each of n -1 of the n playing fields and once on the remaining field. Such a schedule is called a 'balanced tournament design'.

Then each edge {i, j} represents the match between the teams i and j; if {i, j} is contained in the factor Fk , this match will be played on the k-th day. ) If there are no additional conditions on the schedule, we can use any factorization. At the end of this section we will make a few comments on how to set up 'balanced' schedules. Of course, the above method can also be used to set up a schedule for a league (like. for example, the German soccer league), if we consider the two rounds as two separate tournaments.

For the converse we refer to the sources given above. There is yet another interesting characterization of planarity. If we identify two adjacent vertices u and v in a graph G, we get an elementary contraction of G; more precisely, we omit u and v and replace them by a new vertex w which is adjacent to all vertices which were adjacent to u or v before. 11 shows a subdivision and a contraction of K 3 ,3. A graph G is called contractible to a graph H if H arises from G by a sequence of elementary contractions.

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