Alpha labelings of disjoint union of hairy cycles (Q6643643)

Let \(G\) be a bipartite graph on \(m\) edges and \(\phi: V(G)\to \{0,1,\dots,m\}\) an injective function. The function \(\phi\) is called an \(\alpha\)-labeling of \(G\) if\N\begin{itemize}\N\item[(1)] \(\{|\phi(x)-\phi(y)|: xy\in E(G)\}=\{1,2,\dots,m\}\) and\N\item[(2)] there exists \(\ell\in\{0,1,\dots,m-1\}\) such that if \(xy\in E(G)\) and \(\phi(x)<\phi(y)\), then \(\phi(x)\leq\ell<\phi(y)\).\N\end{itemize}\N\NLet \(C^{S_k}_m\) be the graph arising from \(C_m\) by adding \(k\) pendant edges to every vertex of \(C_m\). Denote by \(nH\) the disjoint union of \(n\) copies of a graph \(H\).\N\NIt is proved that the following graphs admit an \(\alpha\)-labeling:\N\begin{itemize}\N\item[(a)] \(nC^{S_1}_4\), \(n\geq2\),\N\item[(b)] \(2C^{S_k}_4\), \(k\geq1\),\N\item[(c)] \(2C^{S_1}_{4m}\), \(m\geq1\),\N\item[(d)] \(C^{S_1}_{4m}\cup C^{S_1}_{4m-2}\), \(m\geq2\), and\N\item[(e)] \(2C^{S_1}_{4m+2}\), \(m\geq1\).\N\end{itemize}