The total disjoint irregularity strength of some certain graphs

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>