On the Turán Number of Theta Graphs

The Turán number ex ( n , H ) is the maximum number of edges in any graph of order n that contains no copy of H as a subgraph. For any three positive integers p , q , r with p≤ q≤ r and q≥ 2 , let θ (p,q,r) denote the graph obtained from three internally disjoint paths with the same pair of endpoints, where the three paths are of lengths p , q , r , respectively. Let k=p+q+r-1 . In this paper, we obtain the exact value of ex(n,θ (p,q,r)) and characterize the unique extremal graph for n≥ 9k^2-3k and any p , q , r with different parities. This extends a known result on odd cycles.