{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T23:14:46Z","timestamp":1723245286577},"reference-count":14,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T00:00:00Z","timestamp":1583971200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004686","name":"Deanship of Scientific Research, King Faisal University","doi-asserted-by":"publisher","award":["186236","11871192","2019YX002A"],"id":[{"id":"10.13039\/501100004686","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,3,12]]},"abstract":"A spanning path in a graph G<\/jats:italic> is called a Hamiltonian path. To determine which graphs possess such paths is an NP-complete problem. A graph G<\/jats:italic> is called Hamiltonian-connected if any two vertices of G<\/jats:italic> are connected by a Hamiltonian path. We consider here the family of Toeplitz graphs. About them, it is known only for n<\/mml:mi>=<\/mml:mo>3<\/mml:mn><\/mml:mrow><\/mml:math> that T<\/mml:mi><\/mml:mrow>n<\/mml:mi><\/mml:mrow><\/mml:msub>p<\/mml:mi>,<\/mml:mo>q<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math> is Hamiltonian-connected, while some particular cases of T<\/mml:mi><\/mml:mrow>n<\/mml:mi><\/mml:mrow><\/mml:msub>p<\/mml:mi>,<\/mml:mo>q<\/mml:mi>,<\/mml:mo>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math> for p<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math> and q<\/mml:mi>=<\/mml:mo>2,3,4<\/mml:mn><\/mml:mrow><\/mml:math> have also been investigated regarding Hamiltonian connectedness. Here, we prove that the nonbipartite Toeplitz graph T<\/mml:mi><\/mml:mrow>n<\/mml:mi><\/mml:mrow><\/mml:msub>1<\/mml:mn>,<\/mml:mo>q<\/mml:mi>,<\/mml:mo>r<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math> is Hamiltonian-connected for all 1<\/mml:mn><<\/mml:mo>q<\/mml:mi><<\/mml:mo>r<\/mml:mi><<\/mml:mo>n<\/mml:mi><\/mml:mrow><\/mml:math> and n<\/mml:mi>\u2265<\/mml:mo>5<\/mml:mn>r<\/mml:mi>\u2212<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2020\/5608720","type":"journal-article","created":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T23:31:04Z","timestamp":1584055864000},"page":"1-6","source":"Crossref","is-referenced-by-count":4,"title":["The Property of Hamiltonian Connectedness in Toeplitz Graphs"],"prefix":"10.1155","volume":"2020","author":[{"given":"Ayesha","family":"Shabbir","sequence":"first","affiliation":[{"name":"Preparatory Year Deanship, King Faisal University, 31982 Hofuf, Al Ahsa, Saudi Arabia"},{"name":"Department of Mathematics, University of Lahore, Gujrat Campus, Gujrat, Pakistan"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-3175-7191","authenticated-orcid":true,"given":"Muhammad Faisal","family":"Nadeem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS, University Islamabad Lahore Campus, Pakistan"}]},{"given":"Tudor","family":"Zamfirescu","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics, University of Dortmund, 44221 Dortmund, Germany"},{"name":"Romanian Academy, Bucharest, Romania"}]}],"member":"98","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218x(91)90024-q"},{"key":"2","first-page":"23","volume-title":"On strongly Hamiltonian abelian group graphs, Combinatorial Mathematics VIII","volume":"884","year":"1981"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/s0304-3975(00)00230-9"},{"key":"5","first-page":"119","volume-title":"A characterization of infinite, bipartite Toeplitz graphs","volume":"1","year":"1995"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-012-1185-8"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/s0012-365x(02)00685-4"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/s0012-365x(01)00136-4"},{"key":"9","first-page":"303","volume":"116","year":"2014","journal-title":"Ars Combinatoria"},{"key":"10","first-page":"511","volume":"109","year":"2013","journal-title":"Ars Combinatoria"},{"issue":"2","key":"11","first-page":"145","volume":"53","year":"2010","journal-title":"Bulletin Math\u00e9matique de la Soci\u00e9t\u00e9 des Sciences Math\u00e9matiques de Roumanie"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0859-0_8"},{"key":"13","first-page":"21","volume":"42","year":"1963","journal-title":"Journal de Math\u00e9matiques Pures et Appliqu\u00e9es"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365x(95)00111-9"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190100103"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/5608720.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/5608720.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2020\/5608720.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,12]],"date-time":"2020-03-12T23:31:08Z","timestamp":1584055868000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/complexity\/2020\/5608720\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,12]]},"references-count":14,"alternative-id":["5608720","5608720"],"URL":"https:\/\/doi.org\/10.1155\/2020\/5608720","relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,12]]}}}