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Total vertex irregularity strength of disjoint union of Helm graphs. (English) Zbl 1257.05144

Summary: A total vertex irregular \(k\)-labeling \(\varphi\) of a graph \(G\) is a labeling of the vertices and edges of \(G\) with labels from the set \(\{1,2,\dots, k\}\) in such a way that for any two different vertices \(x\) and \(y\) their weights \(wt(x)\) and \(wt(y)\) are distinct.
Here, the weight of a vertex \(x\) in \(G\) is the sum of the label of \(x\) and the labels of all edges incident with the vertex \(x\). The minimum \(k\) for which the graph \(G\) has a vertex irregular total \(k\)-labeling is called the total vertex irregularity strength of \(G\). We have determined an exact value of the total vertex irregularity strength of disjoint union of Helm graphs.

MSC:

05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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