Turán number and decomposition number of intersecting odd cycles

Given a graph H, the Turán function ex(n,H) is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. Let s,t be integers and let Hs,t be a graph consisting of s triangles and t cycles of odd lengths at least 5 which intersect in exactly one common vertex. Erdős et al. (1995) determined the Turán function ex(n,Hs,0) and the corresponding extremal graphs. Recently, Hou et al. (2016) determined ex(n,H0,t) and the extremal graphs, where the t cycles have the same odd length q with q⩾5. In this paper, we further determine ex(n,Hs,t) and the extremal graphs, where s⩾0 and t⩾1. Let ϕ(n,H) be the smallest integer such that, for all graphs G on n vertices, the edge set E(G) can be partitioned into at most ϕ(n,H) parts, of which every part either is a single edge or forms a graph isomorphic to H. Pikhurko and Sousa conjectured that ϕ(n,H)=ex(n,H) for χ(H)⩾3 and all sufficiently large n. Liu and Sousa (2015) verified the conjecture for Hs,0. In this paper, we further verify Pikhurko and Sousa's conjecture for Hs,t with s⩾0 and t⩾1.