{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T02:51:56Z","timestamp":1769914316137,"version":"3.49.0"},"reference-count":19,"publisher":"Wiley","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["MOST 103-2115-M-029-001"],"award-info":[{"award-number":["MOST 103-2115-M-029-001"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>An<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-<jats:italic>vertex-antimagic edge labeling<\/jats:italic>(or an<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-<jats:italic>VAE<\/jats:italic>labeling, for short) of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>is a bijective mapping from the edge set<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>E<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of a graph<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>to the set of integers<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mn>1,2<\/mml:mn><mml:mo>,<\/mml:mo><mml:mo>\u2026<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo stretchy=\"false\">|<\/mml:mo><mml:mi>E<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">|<\/mml:mo><\/mml:math>with the property that the vertex-weights form an arithmetic sequence starting from<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:math>and having common difference<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>are two positive integers, and the vertex-weight is the sum of the labels of all edges incident to the vertex. A graph is called<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-antimagic if it admits an<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-<jats:italic>VAE<\/jats:italic>labeling. In this paper, we investigate the existence of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-<jats:italic>vertex-antimagic edge deficiency<\/jats:italic>, as an extension of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-VAE labeling, for measuring how close a graph is away from being an<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-antimagic graph. Furthermore, the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.<\/jats:p>","DOI":"10.1155\/2015\/320616","type":"journal-article","created":{"date-parts":[[2015,6,10]],"date-time":"2015-06-10T16:29:03Z","timestamp":1433953743000},"page":"1-7","source":"Crossref","is-referenced-by-count":4,"title":["On<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>-Vertex-Antimagic Edge Labeling of Regular Graphs"],"prefix":"10.1155","volume":"2015","author":[{"given":"Martin","family":"Ba\u010da","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Informatics, Technical University, Letn\u00e1 9, 04200 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrea","family":"Semani\u010dov\u00e1-Fe\u0148ov\u010d\u00edkov\u00e1","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Informatics, Technical University, Letn\u00e1 9, 04200 Ko\u0161ice, Slovakia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3432-8579","authenticated-orcid":true,"given":"Tao-Ming","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Tunghai University, Taichung 40704, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guang-Hui","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"9","year":"1996"},{"key":"10","volume":"16, article ds6","year":"2013","journal-title":"The Electronic Journal of Combinatorics"},{"key":"11","year":"1990"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20027"},{"key":"5","volume-title":"Arithmetisch antimagische Graphen","year":"1993"},{"key":"6","first-page":"129","volume":"42","year":"1996","journal-title":"Ars Combinatoria"},{"key":"7","first-page":"33","volume":"46","year":"1997","journal-title":"Ars Combinatoria"},{"key":"8","first-page":"85","volume":"17","year":"1998","journal-title":"Mitteilungen der Mathematischen Gesellschaft in Hamburg"},{"key":"4","first-page":"297","volume":"48","year":"1998","journal-title":"Ars Combinatoria"},{"key":"3","first-page":"217","volume":"35","year":"2000","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"14","first-page":"251","volume":"37","year":"2001","journal-title":"Journal of Combinatorial Mathematics and Combinatorial Computing"},{"key":"2","first-page":"237","volume":"58","year":"2000","journal-title":"Utilitas Mathematica"},{"key":"16","first-page":"207","volume":"70","year":"2004","journal-title":"Ars Combinatoria"},{"key":"15","first-page":"3","volume":"61","year":"2002","journal-title":"Utilitas Mathematica"},{"key":"19"},{"issue":"2","key":"13","first-page":"177","volume":"42","year":"2006","journal-title":"SUT Journal of Mathematics"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1007\/bf02392606"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2008.12.020"},{"issue":"6","key":"18","first-page":"935","volume":"34","year":"2003","journal-title":"Indian Journal of Pure and Applied Mathematics"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/320616.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/320616.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/320616.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,8,18]],"date-time":"2016-08-18T13:04:56Z","timestamp":1471525496000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2015\/320616\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"references-count":19,"alternative-id":["320616","320616"],"URL":"https:\/\/doi.org\/10.1155\/2015\/320616","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015]]}}}