Survey of certain valuations of graphs (Q2725185)

For a finite graph \(G=(V,E)\) with \(|V|=v\) and \(|E|=e\), a valuation of \(G\) may be loosely defined as a bijection \(\lambda:V\to\{1,\ldots,v\}\) or \(\lambda:E\to\{1,\ldots,e\}\) or \(\lambda:V\cup E\to\{1,\ldots,v+e\}\) subject to given conditions relative to the incidence relation of \(G\). The present expository article discusses the following three kinds of valuations of graphs and the classes of graphs that admit them. NEWLINENEWLINENEWLINE\(G\) is called antimagic if it admits a valuation of \(E\) such that the sums of the labels of the \(v\) different vertex-cocycles are all distinct. An edge-magic total labeling is a valuation \(\lambda\) of \(V\cup E\) such that for all \([x,y]\in E,\;\lambda(x)+\lambda(y)+\lambda([x,y])\) equals some given constant. Finally, a vertex-magic total labeling is a valuation \(\lambda\) of \(V\cup E\) such that for all \(x\in V,\;\lambda(x)+\sum_{[x,y]\in E}\lambda([x,y])\) equals some given constant.