On the Turan Number of Theta Graphs

The Turan 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 theta(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,theta(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.