## Abstract

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* gem_{4} and gem_{5}. The graph gem_{4} consists of the path *P*
_{4} 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 gem_{5} is similarly defined with the path *P*
_{5} on five vertices. We determine the Turán functions ex(*n,* gem_{4}) and ex(*n,* gem_{5}), and verify the conjecture of Pikhurko and Sousa when *H* is the graph gem_{4} and gem_{5}.