Tournaments
Task number: 4517
Variant
Each team played exactly five teams in their category?Solution
For the category \(A\) such a plan is not possible, because 5 edges would come out of each of the 21 vertices. Each edge has two ends, so \(\frac{21{\cdot} 5}2\) would correspond to the number of edges, which is not an integer.
For the \(B\) category, a tournament plan can be designed, there are many possibilities. For example, one can take a disjunctive union of two \(K_6\) or a graph of the icosahedron, or number the vertices \({0,\dots,11}\) and join each \(i\) with \(i-2,i-1,i+1,i+2\) and \(i+6 \pmod{12}\), see figure.
Variant
Each team in the \(A\) category played exactly four teams in the \(B\) category, and all teams in the \(B\) category played the same number of teams in the \(A\) category (i.e., is there some number of \(b\) such that each team in the \(B\) category plays \(b\) teams)?Solution
In this case, the tournament schedule would correspond to a bipartite graph, with teams in category \(A\) forming one side and teams in category \(B\) forming the other.
In each vertex of the set \(A\), 4 edges come out of the set \(A\), so there are 84 edges in total. This must correspond to \(b=12\) edges leading to \(B\), and from there \(b=7\).
Such a graph can also be constructed, e.g. as a disjunctive union of three \(K_{7{,}4}\).
