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

#### Keywords

- gem graph
- Turán function
- extremal graph
- graph decomposition

#### MSC 2010

- 05C35
- 05C70

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