Totally antimagic total labeling of helm and gear graphs

Source abstract

<p>A total labeling of a graph <em>G</em> is a bijection from the union of the vertex set and the edge set of <em>G</em> to the set {1,2,...,|<em>V</em>(<em>G</em>)|+|<em>E</em>(<em>G</em>)|}. Under a total labeling, the vertex-weight of a vertex is defined as the sum of its label and the labels of all edges incident to it. Similarly, the edge-weight of an edge is the sum of its label and the labels of its two end vertices. A total labeling is said to be edge-antimagic total if all the edge-weights are pairwise distinct, and vertex-antimagic total if all the vertex-weights are pairwise distinct. If a total labeling is edge-antimagic total and vertex-antimagic total at the same time, then it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we show that helm graphs <em>H</em><sub>n</sub> and gear graphs <em>G</em><sub>n</sub> are totally antimagic total graphs.</p>