{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:19:48Z","timestamp":1760239188526,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,21]],"date-time":"2020-10-21T00:00:00Z","timestamp":1603238400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006261","name":"Taif University","doi-asserted-by":"publisher","award":["TURSP-2020\/20"],"award-info":[{"award-number":["TURSP-2020\/20"]}],"id":[{"id":"10.13039\/501100006261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we generalize the orthogonal double covers (ODC) of Kn,n as follows. The circular intensely orthogonal double cover design (CIODCD) of X=Kn,n,\u2026,n\ufe38m is defined as a collection T={G00,G10,\u2026,G(n\u22121)0}\u222a{G01,G11,\u2026,G(n\u22121)1} of isomorphic spanning subgraphs of X such that every edge of X appears twice in the collection T,E(Gi0)\u2229E(Gj0)=E(Gi1)\u2229E(Gj1)=0,i\u2260jand\u00a0E(Gi0)\u2229E(Gj1)=\u03bb=m2,i,j\u2208\u2124n. We define the half starters and the symmetric starters matrices as constructing methods for the CIODCD of X. Then, we introduce some results as a direct application to the construction of CIODCD of X by the symmetric starters matrices.<\/jats:p>","DOI":"10.3390\/sym12101743","type":"journal-article","created":{"date-parts":[[2020,10,23]],"date-time":"2020-10-23T02:01:42Z","timestamp":1603418502000},"page":"1743","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Circular Intensely Orthogonal Double Cover Design of Balanced Complete Multipartite Graphs"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2575-5103","authenticated-orcid":false,"given":"M.","family":"Higazy","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"},{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2009-3905","authenticated-orcid":false,"given":"A.","family":"El-Mesady","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2757-2765","authenticated-orcid":false,"given":"Emad E.","family":"Mahmoud","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7658-7085","authenticated-orcid":false,"given":"Monagi H.","family":"Alkinani","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Artificial Intelligence, College of Computer Sciences and Engineering, University of Jeddah, Jeddah 21493, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/S0167-5060(08)70704-0","article-title":"Block designs with cyclic block structure","volume":"6","author":"Hering","year":"1980","journal-title":"Ann. 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