{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:10Z","timestamp":1740134530940,"version":"3.37.3"},"reference-count":15,"publisher":"Wiley","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":[{"DOI":"10.13039\/501100011093","name":"University of Phayao, Thailand","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100011093","id-type":"DOI","asserted-by":"crossref"}]}],"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 and \u03d5<\/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>k<\/mml:mi>}<\/mml:mo><\/mml:math> be a k<\/mml:mi><\/mml:mrow><\/mml:math>-total coloring. Let w<\/mml:mi>(<\/mml:mo>v<\/mml:mi>)<\/mml:mo><\/mml:math> denote the sum of color on a vertex v<\/mml:mi><\/mml:mrow><\/mml:math> and colors assigned to edges incident to v<\/mml:mi><\/mml:mrow><\/mml:math>. If w<\/mml:mi>(<\/mml:mo>u<\/mml:mi>)<\/mml:mo>\u2260<\/mml:mo>w<\/mml:mi>(<\/mml:mo>v<\/mml:mi>)<\/mml:mo><\/mml:math> whenever u<\/mml:mi>v<\/mml:mi>\u2208<\/mml:mo>E<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>, then \u03d5<\/mml:mi><\/mml:mrow><\/mml:math> is called a neighbor sum distinguishing total coloring. The smallest integer k<\/mml:mi><\/mml:mrow><\/mml:math> such that G<\/mml:mi><\/mml:mrow><\/mml:math> has a neighbor sum distinguishing k<\/mml:mi><\/mml:mrow><\/mml:math>-total coloring is denoted by t<\/mml:mi>n<\/mml:mi>d<\/mml:mi>i<\/mml:mi><\/mml:mrow>\u2211<\/mml:mo>\u200a<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:msub>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>. In 2014, Dong and Wang obtained the results about t<\/mml:mi>n<\/mml:mi>d<\/mml:mi>i<\/mml:mi><\/mml:mrow>\u2211<\/mml:mo>\u200a<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:msub>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math> depending on the value of maximum average degree. A k<\/mml:mi><\/mml:mrow><\/mml:math>-assignment L<\/mml:mi><\/mml:mrow><\/mml:math> of G<\/mml:mi><\/mml:mrow><\/mml:math> is a list assignment L<\/mml:mi><\/mml:mrow><\/mml:math> of integers to vertices and edges with L<\/mml:mi>(<\/mml:mo>v<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mfenced>=<\/mml:mo>k<\/mml:mi><\/mml:math> for each vertex v<\/mml:mi><\/mml:mrow><\/mml:math> and L<\/mml:mi>(<\/mml:mo>e<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mfenced>=<\/mml:mo>k<\/mml:mi><\/mml:math> for each edge e<\/mml:mi><\/mml:mrow><\/mml:math>. A total-<\/jats:italic>L<\/mml:mi><\/mml:mrow><\/mml:math>-coloring<\/jats:italic> is a total coloring \u03d5<\/mml:mi><\/mml:mrow><\/mml:math> of G<\/mml:mi><\/mml:mrow><\/mml:math> such that \u03d5<\/mml:mi>(<\/mml:mo>v<\/mml:mi>)<\/mml:mo>\u2208<\/mml:mo>L<\/mml:mi>(<\/mml:mo>v<\/mml:mi>)<\/mml:mo><\/mml:math> whenever v<\/mml:mi>\u2208<\/mml:mo>V<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math> and \u03d5<\/mml:mi>(<\/mml:mo>e<\/mml:mi>)<\/mml:mo>\u2208<\/mml:mo>L<\/mml:mi>(<\/mml:mo>e<\/mml:mi>)<\/mml:mo><\/mml:math> whenever e<\/mml:mi>\u2208<\/mml:mo>E<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>. We state that G<\/mml:mi><\/mml:mrow><\/mml:math> has a neighbor sum distinguishing total-<\/jats:italic>L<\/mml:mi><\/mml:mrow><\/mml:math>-coloring<\/jats:italic> if G<\/mml:mi><\/mml:mrow><\/mml:math> has a total-L<\/mml:mi><\/mml:mrow><\/mml:math>-coloring such that w<\/mml:mi>(<\/mml:mo>u<\/mml:mi>)<\/mml:mo>\u2260<\/mml:mo>w<\/mml:mi>(<\/mml:mo>v<\/mml:mi>)<\/mml:mo><\/mml:math> for all u<\/mml:mi>v<\/mml:mi>\u2208<\/mml:mo>E<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>. The smallest integer k<\/mml:mi><\/mml:mrow><\/mml:math> such that G<\/mml:mi><\/mml:mrow><\/mml:math> has a neighbor sum distinguishing total-L<\/mml:mi><\/mml:mrow><\/mml:math>-coloring for every k<\/mml:mi><\/mml:mrow><\/mml:math>-assignment L<\/mml:mi><\/mml:mrow><\/mml:math> is denoted by C<\/mml:mi>h<\/mml:mi><\/mml:mrow>\u2211<\/mml:mo>\u200a<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow>\u2032<\/mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>. In this paper, we strengthen results by Dong and Wang by giving analogous results for C<\/mml:mi>h<\/mml:mi><\/mml:mrow>\u2211<\/mml:mo>\u200a<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow>\u2032<\/mml:mi>\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2017\/5897049","type":"journal-article","created":{"date-parts":[[2017,11,7]],"date-time":"2017-11-07T23:31:16Z","timestamp":1510097476000},"page":"1-4","source":"Crossref","is-referenced-by-count":0,"title":["Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers"],"prefix":"10.1155","volume":"2017","author":[{"given":"Patcharapan","family":"Jumnongnit","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0421-3631","authenticated-orcid":true,"given":"Kittikorn","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"17","doi-asserted-by":"publisher","DOI":"10.1360\/03YS0207"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2007.07.091"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1007\/s10878-012-9527-2"},{"issue":"12","key":"15","first-page":"1462","volume":"39","year":"2009","journal-title":"Sci. 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