On certain vertex valuations of a graph. I (Q1184911)

A graph \(G=(V,E)\) is said to be additively \((k,d)\)-arithmetic if \(G\) admits a vertex function \(f\) such that \(\{k,k+d,k+2d,\dots\}\) is the set of values of the induced edge function \(f_ e\) where \(f_ e(uv)=f(u)+f(v)\) for each edge \(uv\in E\). Several properties of such graphs are established and a connection with certain graceful graphs is shown.