Turán function and H-decomposition problem for gem graphs.

Given a graph H, the Turcin 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 phi(n, H) be the smallest number phi such that any graph G of order n admits an H-decomposition with at most phi parts. Pikhurko and Sousa conjectured that phi(n, H) = ex(n, H) for (chi) double under dot (H) >= 3 and all sufficiently large n. Their conjecture has been verified by Ozkahya 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 Turan 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 gems.