An edge labeling of graphs from Rado’s partition regularity condition

A vertex \(v\) is called an AR-vertex, if \(v\) has distinct sum of edge labels for each distinct subset of edges incident on \(v\). i.e., if \(\{x_1,x_2,\dots,x_k\}\) are the edge labels of the edges incident on \(v\), then the \(2^k\) subset sums are all distinct. An injective edge labeling \(f\) of a graph \(G\) is said to be an AR-labeling of \(G\) if \(f:E \rightarrow \mathbb{N}\) is such that every vertex in \(G\) is an AR-vertex under \(f\). A graph \(G\) is said to be an AR-graph if there exists an AR-labeling \(f:E\rightarrow \{1,2,\dots,m\}\), where \(m\) denotes the number of edges of \(G\). A study of AR-labeling and AR-graphs is initiated in this paper.