{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T11:55:46Z","timestamp":1774439746103,"version":"3.50.1"},"reference-count":5,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>be a graph on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>vertices and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>\ud835\udca2<\/mml:mi><\/mml:mrow><\/mml:math>a collection of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>subgraphs of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>, one for each vertex, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>\ud835\udca2<\/mml:mi><\/mml:mrow><\/mml:math>is an orthogonal double cover (ODC) of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>if every edge of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>occurs in exactly two members of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi>\ud835\udca2<\/mml:mi><\/mml:mrow><\/mml:math>and any two members share an edge whenever the corresponding vertices are adjacent in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>and share no edges whenever the corresponding vertices are nonadjacent in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>H<\/mml:mi><\/mml:mrow><\/mml:math>. In this paper, we are concerned with the Cartesian product of symmetric starter vectors of orthogonal double covers of the complete bipartite graphs and using this method to construct ODCs by new disjoint unions of complete bipartite graphs.<\/jats:p>","DOI":"10.1155\/2013\/265136","type":"journal-article","created":{"date-parts":[[2013,4,16]],"date-time":"2013-04-16T17:07:15Z","timestamp":1366132035000},"page":"1-4","source":"Crossref","is-referenced-by-count":4,"title":["On Cartesian Products of Orthogonal Double Covers"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3377-155X","authenticated-orcid":true,"given":"R.","family":"El Shanawany","sequence":"first","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufiya University, Menouf 32952, Egypt"}]},{"given":"M.","family":"Higazy","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufiya University, Menouf 32952, Egypt"}]},{"given":"A.","family":"El Mesady","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufiya University, Menouf 32952, Egypt"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1016546402248"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(03)00274-9"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(03)00269-5"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2009.06.005"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4529-6"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/265136.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/265136.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2013\/265136.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,30]],"date-time":"2016-07-30T08:35:23Z","timestamp":1469867723000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2013\/265136\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":5,"alternative-id":["265136","265136"],"URL":"https:\/\/doi.org\/10.1155\/2013\/265136","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013]]}}}