Turán function and h-decomposition problem for gem graphs

Given a graph H, the Turán function ex(n, H) is the maximum number of edges in a graph on n vertices not containing H as a subgraph. For two graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let Φ(n, H) be the smallest number Φ such that any graph G of order n admits an H-decomposition with at most Φ parts. Pikhurko and Sousa conjectured that Φ(n, H) = ex(n, H) for χ(H) ≥ 3 and all sufficiently large n. Their conjecture has been verified by Özkahya and Person for all edge-critical graphs H. In this article, we consider the gem graphs gem₄ and gem₅. The graph gem₄ consists of the path P₄ with four vertices a, b, c, d and edges ab, bc, cd plus a universal vertex u adjacent to a, b, c, d, and the graph gem₅ is similarly defined with the path P₅ on five vertices. We determine the Turán functions ex(n, gem₄) and ex(n, gem₅), and verify the conjecture of Pikhurko and Sousa when H is the graph gem₄ and gem₅