All 5-stars are Skolem graceful

A graph G = (V, E) is Skolem graceful if its vertices can be labelled 1, 2, ... | V | so that the edges are labelled 1, 2, ... | E | where each edge label is the absolute difference of the labeLs of the two end vertices. A k-star is a graph with k components, with almost one vertex of degree exceeding 1 in each component. It is known that all 1-stars, 4-stars are Skolem graceful and that not all 2-stars and 3-stars are Skolem graceful. In this paper we prove that all 5-stars are Skolem graceful, and obtain a necessary condition for a k-star to be Skolem graceful.