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A new upper bound for the total vertex irregularity strength of graphs. (English) Zbl 1210.05117
Summary: We investigate the following modification of the well-known irregularity strength of graphs. Given a total weighting \(w\) of a graph \(G=(V,E)\) with elements of a set \(\{1,2,\dots ,s\}\), denote \(wt_G(v)=\sum _{e\ni v}w(e)+w(v)\) for each \(v\in V\). The smallest \(s\) for which exists such a weighting with \(wt_G(u)\neq wt_G(v)\) whenever \(u\) and \(v\) are distinct vertices of \(G\) is called the total vertex irregularity strength of this graph, and is denoted by tvs\((G)\). We prove that tvs\((G)\leq 3\lceil n/\delta \rceil\) for each graph of order \(n\) and with minimum degree \(\delta >0\).

MSC:
05C75 Structural characterization of families of graphs
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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