Let $G=(V,E)$ be a graph with minimal degree $\delta(G) \geq 2$. The set of vertices $V(G)$ can be written as a union of cycles. 


Question: is there a universal constant $k\in \mathbb{N}$ such that every graph $G$ with $\delta(G) \geq 2$ can be written as a union of a collection ${\cal C}$ of cycles (not necessarily induced subgraphs, just subgraphs) such that whenever $C\_1\neq C\_2\in {\cal C}$ we have $|C\_1\cap C\_2| \leq k$?



Union of cycles
Counterexample: Consider a graph consisting of three paths $u a\_1\cdots a\_{k-1} v$, $u b\_1\cdots b\_{k-1} v$, and $u c\_1\cdots c\_{k-1} v$, and no other vertices or edges. Every cycle contains exactly two of those paths so any two cycles share $k+1$ vertices.


