Scheduling Partial Round Robin Tournaments Subject to Home Away Pattern Sets

We consider the following sports scheduling problem. Consider 2n teams in a sport league. Each pair of teams must play exactly one match in 2n− 1 days. That is, n games are held simultaneously in a day. We want to make a schedule which has n(2n − 1) games for 2n − 1 days. When we make a schedule, the schedule must satisfy a constraint according to the HAP set, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for an HAP set to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilbert basis of such a monoid, the problem is naturally generalized to a scheduling problem for not all pairs of teams described by a regular graph. In this paper, we show a regular graph such that the corresponding linear inequalities have a solution but do not have any integral solution. Moreover we discuss for which regular graphs the statement generalizing the conjecture holds.