Sports scheduling for not all pairs of teams

TL;DR

This paper investigates a sports scheduling problem involving 2n teams, focusing on conditions for feasible schedules with specific home-away constraints, and explores the conjecture relating linear and integral solutions using perfect matchings.

Contribution

It reformulates Briskorn's conjecture using perfect matchings and monoids, and provides counterexamples and conditions for the conjecture's validity in regular graphs.

Findings

Counterexample of a regular graph with a linear solution but no integral solution

Reformulation of the scheduling problem using perfect matchings and monoids

Discussion of conditions under which the conjecture holds

Abstract

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.