We introduce the concept of a free \(a\)-valuation of a graph, and prove that the vertex-disjoint union of any collection of graphs with free \(\alpha\)-valuations has an \(\alpha\)-valuation. Many bipartite graphs have free \(\alpha\)-valuations, including the complete bipartite graph \(K_{m,n}\) when \(m > 1\) and \(n > 2\), and the \(d\)-cube \(Q_d\) for \(d > 2\).