{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,13]],"date-time":"2025-09-13T16:09:09Z","timestamp":1757779749958,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A binary (cyclic) Gray code\u00a0is a (cyclic) ordering of all binary strings of the same length such that any two consecutive\u00a0strings differ in a single bit. This corresponds to a Hamiltonian path (cycle) in the hypercube.\u00a0Fink showed that every perfect matching in the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle, confirming a conjecture of Kreweras.\u00a0In this paper, we study the \"path version\" of this problem. Namely, we characterize when a perfect matching in $Q_n$ extends to\u00a0a Hamiltonian path between two prescribed vertices of opposite parity. Furthermore, we characterize when a perfect matching in $Q_n$ with two faulty vertices extends to a Hamiltonian cycle. In both cases we show that for all dimensions $n\\ge 5$ the only forbidden configurations are so-called half-layers, which are certain natural obstacles. These results thus extend Kreweras' conjecture with an additional edge, or with two faulty vertices. The proof for the case $n=5$ is computer-assisted.<\/jats:p>","DOI":"10.37236\/6928","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:23:51Z","timestamp":1578669831000},"source":"Crossref","is-referenced-by-count":3,"title":["Extending Perfect Matchings to Gray Codes with Prescribed Ends"],"prefix":"10.37236","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3608-2533","authenticated-orcid":false,"given":"Petr","family":"Gregor","sequence":"first","affiliation":[]},{"given":"Tom\u00e1\u0161","family":"Novotn\u00fd","sequence":"additional","affiliation":[]},{"given":"Riste","family":"\u0160krekovski","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,6,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p56\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p56\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:31:14Z","timestamp":1579235474000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i2p56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,22]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/6928","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,6,22]]},"article-number":"P2.56"}}