  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 table, 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 a HAP table 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.
