Edge-magic total labelings of wheels (Q2706949)

For a graph \(G\) with \(v\) vertices and \(e\) edges, a one-to-one mapping \(\lambda\) from \(V(G)\cup E(G)\) to the set \(\{1,2,\dots, v+e\}\) is an edge-magic total labelling (EMTL) if for every edge \(xy\), \(\lambda(x)+ \lambda(y)+ \lambda(xy)= k\), for some constant \(k\). A necessary condition for the \(n\)-spoke wheel \(W_n\) to admit an EMTL is \(n\not\equiv 3\pmod 4\). In this paper it is shown, by direct construction, that \(W_n\) has an EMTL whenever \(n\equiv 0\) or \(1\pmod 4\).