{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,17]],"date-time":"2024-04-17T05:13:21Z","timestamp":1713330801548},"reference-count":6,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Directorate General of Higher Education, Indonesia","award":["018\/SP2H\/P\/K7\/KM\/2016"]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2017]]},"abstract":"Let G<\/mml:mi><\/mml:mrow><\/mml:math> be a graph of order v<\/mml:mi><\/mml:mrow><\/mml:math> and size e<\/mml:mi><\/mml:mrow><\/mml:math>. An edge-magic labeling<\/jats:italic> of G<\/mml:mi><\/mml:mrow><\/mml:math> is a bijection f<\/mml:mi>:<\/mml:mo>V<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo>\u222a<\/mml:mo>E<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo>\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:mo>e<\/mml:mi>}<\/mml:mo><\/mml:math> such that f<\/mml:mi>(<\/mml:mo>x<\/mml:mi>)<\/mml:mo>+<\/mml:mo>f<\/mml:mi>(<\/mml:mo>x<\/mml:mi>y<\/mml:mi>)<\/mml:mo>+<\/mml:mo>f<\/mml:mi>(<\/mml:mo>y<\/mml:mi>)<\/mml:mo><\/mml:math> is a constant for every edge x<\/mml:mi>y<\/mml:mi>\u2208<\/mml:mo>E<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>. An edge-magic labeling f<\/mml:mi><\/mml:mrow><\/mml:math> of G<\/mml:mi><\/mml:mrow><\/mml:math> with f<\/mml:mi>(<\/mml:mo>V<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo>)<\/mml:mo>=<\/mml:mo>{<\/mml:mo>1,2<\/mml:mn>,<\/mml:mo>3<\/mml:mn>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>v<\/mml:mi>}<\/mml:mo><\/mml:math> is called a super edge-magic labeling<\/jats:italic>. Furthermore, the edge-magic deficiency<\/jats:italic> of a graph G<\/mml:mi><\/mml:mrow><\/mml:math>, \u03bc<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>, is defined as the smallest nonnegative integer n<\/mml:mi><\/mml:mrow><\/mml:math> such that G<\/mml:mi>\u222a<\/mml:mo>n<\/mml:mi>K<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math> has an edge-magic labeling. Similarly, the super edge-magic deficiency<\/jats:italic> of a graph G<\/mml:mi><\/mml:mrow><\/mml:math>, \u03bc<\/mml:mi><\/mml:mrow>s<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>, is either the smallest nonnegative integer n<\/mml:mi><\/mml:mrow><\/mml:math> such that G<\/mml:mi>\u222a<\/mml:mo>n<\/mml:mi>K<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math> has a super edge-magic labeling or +<\/mml:mo>\u221e<\/mml:mi><\/mml:math> if there exists no such integer n<\/mml:mi><\/mml:mrow><\/mml:math>. In this paper, we investigate the (super) edge-magic deficiency of chain graphs. Referring to these, we propose some open problems.<\/jats:p>","DOI":"10.1155\/2017\/5156974","type":"journal-article","created":{"date-parts":[[2017,7,12]],"date-time":"2017-07-12T21:03:41Z","timestamp":1499893421000},"page":"1-6","source":"Crossref","is-referenced-by-count":2,"title":["New Results on the (Super) Edge-Magic Deficiency of Chain Graphs"],"prefix":"10.1155","volume":"2017","author":[{"given":"Ngurah","family":"Anak Agung Gede","sequence":"first","affiliation":[{"name":"Department of Civil Engineering, Universitas Merdeka Malang, Jl. Taman Agung No. 1, Malang 65146, Indonesia"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-3518-7587","authenticated-orcid":true,"family":"Adiwijaya","sequence":"additional","affiliation":[{"name":"School of Computing, Telkom University, Jl. Telekomunikasi No. 1, Bandung 40257, Indonesia"}]}],"member":"98","reference":[{"key":"5","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1970-084-1"},{"issue":"2","key":"1","first-page":"105","volume":"34","year":"1998","journal-title":"SUT Journal of Mathematics"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/S1571-0653(04)00074-5"},{"key":"6","first-page":"3","volume":"40","year":"2008","journal-title":"Australasian Journal of Combinatorics"},{"key":"4","volume":"16","year":"2015","journal-title":"Electron. J. 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