A $2$-cycle on a graph $G=(V,E)$ is a function $d: E\times E\to \mathbb{Z}$ such that for each edge $e$, both $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$. For an oriented cycle $C$ and an edge $e$ of $G$, define $C(e)=+1$ if $C$ traverses $e$ in forward direction, and $C(e)=-1$ if $C$ traverses $e$ in backward direction. Then examples of $2$-cycles are: take two vertex-disjoint oriented cycles $C$ and $D$ of $G$ and define $d(e,f) = C(e)D(f)$. Also on each $K_{3,3}$- and $K_5$-subdivision are $2$-cycles. In this talk, we show that each $2$-cycle on $G$ can be written as a sum of four types of special $2$-cycles.

This is joint work with Serguei Norine and Robin Thomas.