{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:30:55Z","timestamp":1762522255243,"version":"3.40.5"},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2019,2,3]],"date-time":"2019-02-03T00:00:00Z","timestamp":1549152000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CGANT University of Jember"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2019,2,3]]},"abstract":"A simple graph G<\/mml:mi>=<\/mml:mo>(<\/mml:mo>V<\/mml:mi>,<\/mml:mo>E<\/mml:mi>)<\/mml:mo><\/mml:math> is said to be an H<\/mml:mi><\/mml:mrow><\/mml:math>-covering if every edge of G<\/mml:mi><\/mml:mrow><\/mml:math> belongs to at least one subgraph isomorphic to H<\/mml:mi><\/mml:mrow><\/mml:math>. A bijection f<\/mml:mi>:<\/mml:mo>V<\/mml:mi>\u222a<\/mml:mo>E<\/mml:mi>\u2192<\/mml:mo>{<\/mml:mo>1,2<\/mml:mn>,<\/mml:mo>3<\/mml:mn>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>V<\/mml:mi><\/mml:mrow><\/mml:mfenced>+<\/mml:mo>E<\/mml:mi><\/mml:mrow><\/mml:mfenced>}<\/mml:mo><\/mml:math> is an (a,d)-H<\/mml:mi><\/mml:mrow><\/mml:math>-antimagic total labeling of G<\/mml:mi><\/mml:mrow><\/mml:math> if, for all subgraphs H<\/mml:mi><\/mml:mrow>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math> isomorphic to H<\/mml:mi><\/mml:mrow><\/mml:math>, the sum of labels of all vertices and edges in H<\/mml:mi><\/mml:mrow>\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math> form an arithmetic sequence {<\/mml:mo>a<\/mml:mi>,<\/mml:mo>a<\/mml:mi>+<\/mml:mo>d<\/mml:mi>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>(<\/mml:mo>k<\/mml:mi>-<\/mml:mo>1<\/mml:mn>)<\/mml:mo>d<\/mml:mi>}<\/mml:mo><\/mml:math> where a<\/mml:mi>><\/mml:mo>0<\/mml:mn><\/mml:math>, d<\/mml:mi>\u2265<\/mml:mo>0<\/mml:mn><\/mml:math> are two fixed integers and k<\/mml:mi><\/mml:mrow><\/mml:math> is the number of all subgraphs of G<\/mml:mi><\/mml:mrow><\/mml:math> isomorphic to H<\/mml:mi><\/mml:mrow><\/mml:math>. The labeling f<\/mml:mi><\/mml:mrow><\/mml:math> is called super<\/jats:italic> if the smallest possible labels appear on the vertices. A graph that admits (super) (<\/mml:mo>a<\/mml:mi>,<\/mml:mo>d<\/mml:mi>)<\/mml:mo><\/mml:math>-H<\/mml:mi><\/mml:mrow><\/mml:math>-antimagic total labeling is called (super) (<\/mml:mo>a<\/mml:mi>,<\/mml:mo>d<\/mml:mi>)<\/mml:mo><\/mml:math>-H<\/mml:mi><\/mml:mrow><\/mml:math>-antimagic. For a special d<\/mml:mi>=<\/mml:mo>0<\/mml:mn><\/mml:math>, the (super) (<\/mml:mo>a<\/mml:mi>,<\/mml:mo>0<\/mml:mn>)<\/mml:mo><\/mml:math>-H<\/mml:mi><\/mml:mrow><\/mml:math>-antimagic total labeling is called H<\/mml:mi><\/mml:mrow><\/mml:math>-(super)magic labeling. A graph that admits such a labeling is called H<\/mml:mi><\/mml:mrow><\/mml:math>-(super)magic. The m<\/mml:mi><\/mml:mrow><\/mml:math>-shadow of graph G<\/mml:mi><\/mml:mrow><\/mml:math>, D<\/mml:mi><\/mml:mrow>m<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>, is a graph obtained by taking m<\/mml:mi><\/mml:mrow><\/mml:math> copies of G<\/mml:mi><\/mml:mrow><\/mml:math>, namely, G<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>G<\/mml:mi><\/mml:mrow>2<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>G<\/mml:mi><\/mml:mrow>m<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>, and then joining every vertex u<\/mml:mi><\/mml:mrow><\/mml:math> in G<\/mml:mi><\/mml:mrow>i<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>, i<\/mml:mi>\u2208<\/mml:mo>{<\/mml:mo>1,2<\/mml:mn>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>m<\/mml:mi>-<\/mml:mo>1<\/mml:mn>}<\/mml:mo><\/mml:math>, to the neighbors of the corresponding vertex v<\/mml:mi><\/mml:mrow><\/mml:math> in G<\/mml:mi><\/mml:mrow>i<\/mml:mi>+<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>. In this paper we studied the H<\/mml:mi><\/mml:mrow><\/mml:math>-supermagic labelings of D<\/mml:mi><\/mml:mrow>m<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math> where G<\/mml:mi><\/mml:mrow><\/mml:math> are paths and cycles.<\/jats:p>","DOI":"10.1155\/2019\/8780329","type":"journal-article","created":{"date-parts":[[2019,2,3]],"date-time":"2019-02-03T18:36:26Z","timestamp":1549218986000},"page":"1-7","source":"Crossref","is-referenced-by-count":5,"title":["On H<\/mml:mi><\/mml:mrow><\/mml:math>-Supermagic Labelings of m<\/mml:mi><\/mml:mrow><\/mml:math>-Shadow of Paths and Cycles"],"prefix":"10.1155","volume":"2019","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7508-0760","authenticated-orcid":true,"given":"Ika Hesti","family":"Agustin","sequence":"first","affiliation":[{"name":"CGANT Research Group, University of Jember, Indonesia"},{"name":"Department of Mathematics, University of Jember, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8012-1261","authenticated-orcid":true,"given":"F.","family":"Susanto","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Jember, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0575-3039","authenticated-orcid":true,"family":"Dafik","sequence":"additional","affiliation":[{"name":"CGANT Research Group, University of Jember, Indonesia"},{"name":"Department of Mathematics Education, University of Jember, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6611-6458","authenticated-orcid":true,"given":"R. M.","family":"Prihandini","sequence":"additional","affiliation":[{"name":"CGANT Research Group, University of Jember, Indonesia"},{"name":"Department of Elementary School Teacher Education, University of Jember, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3011-4917","authenticated-orcid":true,"given":"R.","family":"Alfarisi","sequence":"additional","affiliation":[{"name":"CGANT Research Group, University of Jember, Indonesia"},{"name":"Department of Elementary School Teacher Education, University of Jember, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9198-3414","authenticated-orcid":true,"given":"I. W.","family":"Sudarsana","sequence":"additional","affiliation":[{"name":"Combinatorial and Applied Mathematics Research Group, University of Tadulako, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"13","first-page":"4610","volume":"2","year":"2010","journal-title":"International Journal of Engineering Science and Technology"},{"year":"2014","key":"6"},{"key":"8","first-page":"273","volume":"71","year":"2009","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"9","first-page":"127","volume":"57","year":"2013","journal-title":"The Australasian Journal of Combinatorics"},{"volume":"893, article 012042","journal-title":"Journal of Physics: Conference Series","year":"2017","key":"3"},{"key":"4","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.procs.2015.12.082","volume":"74","year":"2015","journal-title":"Procedia Computer Science"},{"key":"5","first-page":"233","volume":"133","year":"2017","journal-title":"Ars Combinatoria"},{"journal-title":"AKCE International Journal of Graphs and Combinatorics","year":"2018","key":"1"},{"volume":"943, article 012060","journal-title":"Journal of Physics: Conference Series","year":"2017","key":"2"},{"key":"12","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.endm.2015.05.003","volume":"48","year":"2015","journal-title":"Electronic Notes in Discrete Mathematics"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/j.akcej.2018.01.017"},{"key":"7","first-page":"43","volume":"55","year":"2005","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"10","first-page":"197","volume":"65","year":"2008","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"11","first-page":"333","volume":"83","year":"2010","journal-title":"Utilitas Mathematica"},{"issue":"21","key":"15","volume":"3","year":"2011","journal-title":"Journal of Mathematics Research"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2019\/8780329.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2019\/8780329.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2019\/8780329.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,2,3]],"date-time":"2019-02-03T18:36:31Z","timestamp":1549218991000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2019\/8780329\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,3]]},"references-count":15,"alternative-id":["8780329","8780329"],"URL":"https:\/\/doi.org\/10.1155\/2019\/8780329","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2019,2,3]]}}}