Decomposition of graph to cycle and cut space

Question

Let $G$ be a graph. I want to show that $E(G)$ is disjoint union $C\cup D$ where $C$ and $D$ belong to cycle and cut space respectively.

Answer 1

Note that the result says that we can partition $G$ into $V_1,V_2$ in such a way that all vertices in $G[V_1]$ and $G[V_2]$ have even degree. A proof of this is given in Combinatiorial Problems and Exercises (Lovász):

If all the vertices have even degree then we're done. Assume that $v$ is a vertex of odd degree with neighbours $N$. Let $G'$ be the graph given by removing $v$ and then adding an edge modulo $2$ between all pairs of elements of $N$.

By the induction hypothesis we can partition $G'$ into $W_1,W_2$.

Without loss of generality we can assume that $|N \cap W_1|$ is even and $|N \cap W_2|$ is odd, so we can set $V_1=W_1 \cup \{v\}$ and $V_2=W_2$ and check that everything has even degree as required.