A note on labeling of graphs (Q1393033)

Let \(G=(V,E)\) be a graph which is not a tree. For an injective function \(g:V\to\{0,\dots,| E| -1\}\) define \(g^*:E\to{\mathbb{N}}\) such that \(g^*(uv)=g(u)+g(v)\) for all edges \(uv\in E\). The graph \(G\) is called sequential if \(g^*(E)\) is a sequence of distinct consecutive integers. Furthermore, for graphs \(G\) and \(H\) denote by \(G\odot H\) the graph obtained as follows: for each vertex \(v\) of \(G\) take a copy of \(H\) and join by an edge the vertex \(v\) with each vertex of \(H\). It is known that \(C_n\odot K_2\) is sequential for all odd \(n\). The authors provide examples of further sequential graphs. The main results of the paper show that the following graphs are sequential: \(C_n\odot P_3\) (\(n\) odd), \(K_2\odot C_n\) (\(n\) odd), \(W_n\odot K_1\), where \(W_n\) is a wheel with \(n\geq 4\) vertices and \(n\) is even.