Tur\'an number and decomposition number of intersecting odd cycles

An extremal graph for a given graph $H$ is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $s,t$ be integers and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ cycles of odd lengths at least 5 which intersect in exactly one common vertex. Erd\H{o}s et al. (1995) determined the extremal graphs for $H_{s,0}$. Recently, Hou et al. (2016) determined the extremal graphs for $H_{0,t}$, where the $t$ cycles have the same odd length $q$ with $q\ge 5$. In this paper, we further determine the extremal graphs for $H_{s,t}$ with $s\ge 0$ and $t\ge 1$. Let $\phi(n,H)$ be the largest integer such that, for all graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $\phi(n, H)$ parts, of which every part either is a single edge or forms a graph isomorphic to $H$. Pikhurko and Sousa conjectured that $\phi(n,H)=\ex(n,H)$ for $\chi(H)\geqs3$ and all sufficiently large $n$. Liu and Sousa (2015) verified the conjecture for $H_{s,0}$. In this paper, we further verify Pikhurko and Sousa's conjecture for $H_{s,t}$ with $s\ge 0$ and $t\ge 1$.