In the proof for the NP-completeness of Edge-colouring (paper), there is an intermediate result used called the parity condition, which is formulated in a lemma.
Quoting from the paper:
Let $G$ be a cubic, 3-edge colored graph and V' \subseteq V(G) a set of vertices in $G$. Let E' \subseteq E(G) be the set of edges of $G$ which connect V' to the remainder of the graph. If the number of edges of color $i$ in E' is $k_i$ ($i = 1, 2, 3$), then
$ k_1 \equiv k_2 \equiv k_3 \pmod 2 $
Proof. If $E_{12}$ is the set of edges of $G$ which are colored with color 1 or 2, the $E_{12}$ consists of a collection of cycles. Thus $E_{12}$ meets E' in an even number of edges, and so $k_1 + k_2 \equiv 0 \pmod 2$. Similarly $k_2 \equiv k_3 \pmod 2$. $\Box$
I'm having trouble getting the intuition for the proof. Why does $E_{12}$ necessarily consist of cycles? And why does that fact lead us that $E_{12}$ necessarily meets E' in an even number of edges?