{"title":"The total disjoint irregularity strength of some certain graphs","authors":"M. Tilukay, A. Salman","doi":"10.19184/IJC.2020.4.2.2","DOIUrl":null,"url":null,"abstract":"<div class=\"page\" title=\"Page 1\"><div class=\"layoutArea\"><div class=\"column\"><p><span>Under a totally irregular total </span><em>k</em><span>-labeling of a graph </span><span><em>G</em> </span><span>= (</span><span><em>V</em>,<em>E</em></span><span>), we found that for some certain graphs, the edge-weight set </span><em>W</em><span>(</span><em>E</em><span>) and the vertex-weight set </span><em>W</em><span>(</span><em>V</em><span>) of </span><span><em>G</em> </span><span>which are induced by </span><span><em>k</em> </span><span>= </span><span>ts</span><span>(</span><em>G</em><span>), </span><em>W</em><span>(</span><em>E</em><span>) </span><span>∩ </span><em>W</em><span>(</span><em>V</em><span>) is a non empty set. For which </span><span>k</span><span>, a graph </span><span>G </span><span>has a totally irregular total labeling if </span><em>W</em><span>(</span><em>E</em><span>) </span><span>∩ </span><em>W</em><span>(</span><em>V</em><span>) = </span><span>∅</span><span>? We introduce the total disjoint irregularity strength, denoted by </span><span>ds</span><span>(</span><em>G</em><span>), as the minimum value </span><span><em>k</em> </span><span>where this condition satisfied. We provide the lower bound of </span><span>ds</span><span>(</span><em>G</em><span>) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.</span></p></div></div></div>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"26 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indonesian Journal of Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.19184/IJC.2020.4.2.2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
Under a totally irregular total k-labeling of a graph G= (V,E), we found that for some certain graphs, the edge-weight set W(E) and the vertex-weight set W(V) of Gwhich are induced by k= ts(G), W(E) ∩ W(V) is a non empty set. For which k, a graph G has a totally irregular total labeling if W(E) ∩ W(V) = ∅? We introduce the total disjoint irregularity strength, denoted by ds(G), as the minimum value kwhere this condition satisfied. We provide the lower bound of ds(G) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.
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