{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T14:26:10Z","timestamp":1759847170084,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/144","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:31:58Z","timestamp":1578717118000},"source":"Crossref","is-referenced-by-count":4,"title":["Scheduling Partial Round Robin Tournaments Subject to Home Away Pattern Sets"],"prefix":"10.37236","volume":"16","author":[{"given":"Kenji","family":"Kashiwabara","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,4,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r55\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r55\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:55:45Z","timestamp":1579316145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r55"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,4,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/144","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,4,30]]},"article-number":"R55"}}