Say we have a graph $G$ that has an Eulerian circuit. Break apart its edges into a disjoint union of cycles. Say it can represented as follows: $\{ C_1, C_2, ..., C_n\}$.
Question: Is there a way to traverse $E(G)$ that uses what these cycles are? I'm looking for a sequence/subsequence/pattern thing that uses cycles as I am interested in the way the cycles interact with traversing this graph here.
Below is a picture that might help explain how to do so in context to the problem.
Let's let $W$ represent our eventual walk. Start with $W=C_1$. Now, choose some unused cycle $C_i$ that shares a vertex $v$ with $W$. (This must be possible. If it were not, then the union of all of the used cycles and the union of all of the unused cycles would be in different components, contrary to our assumed assumption that $G$ is connected.) Replace any version of $v$ in $W$ with $C_i$, essentially traversing that cycle in the middle of $W$.
Repeat that process until all of the cycles are used, and $W$ will be an Euler circuit of $G$.
In the context of your graph, let's start with $W=(1,2,3,4,1)$. Now we add in the green cycle, choosing the intersection at $4$. This might give us $W=(1,2,3,\underline{4,3,7,6,5,4},1)$ (where the underlined section is the orientation of the green cycle that we chose). Now we add in the purple cycle intersecting at $1$, which might give us $W=(1,2,3,4,3,7,6,5,4,\underline{1,10,5,1})$. Finally, we add in the blue cycle, which gives the completed Euler circuit of $W=(1,2,3,4,3,7,9,8,7,6,5,4,1,10,5,1)$.