Round-robin tournaments with homogeneous rounds

We study round-robin tournaments for n teams, where in each round a fixed number (g) of teams is present and each team present plays a fixed number (m) of matches in this round. In the tournament each match between two teams is either played once or twice, in the latter case in different rounds. We give necessary combinatorial conditions on the triples (n, g,m) for which such round-robin tournaments can exist, and discuss three general construction methods that concern the cases m = 1, m = 2 and m = g−1. For n ≤ 20 these cases cover 149 of all 173 non-trivial cases that satisfy the necessary conditions. Of these 149 cases a tournament can be constructed in 147 cases. For the remaining 24 cases the tournament does not exist in 2 cases, and is constructed in all other cases. Finally we consider the spreading of rounds for teams, and give some examples where well-spreading is either possible or impossible.