In this paper, we give two constructive proofs that all 4-stars are Skolem-graceful. A 4-star is a graph with 4 components, with at most one vertex of degree exceeding 1 per component. A graph G = (V, E) is Skolem-graceful if its vertices can be labelled 1, 2,..., Absolute value of V so that the edges are labelled 1, 2,..., Absolute value of E, where each edge-label is the absolute difference of the labels of the two end-vertices. Skolem-gracefulness is related to the classic concept of gracefulness, and the methods we develop here may be useful there.
