To find a graph $G$ such that both $G$ and its line graph $L(G)$ are Eulerian, let’s recall the properties of Eulerian graphs:

1. An Eulerian graph is a connected graph in which all vertices have an even degree, and there exists a closed trail (an Eulerian circuit) visiting every edge exactly once.

2. The line graph $L(G)$ of a graph $G$ is constructed such that each vertex in $L(G)$ represents an edge in $G$, and two vertices in $L(G)$ are adjacent if the corresponding edges in $G$ share a common endpoint.

---

Example Construction

Let $G$ be a simple cycle $C_3$ (a triangle):

1. Graph $G$: $G$ is a triangle (3-cycle) with vertices $\{v_1, v_2, v_3\}$ and edges $\{e_1 = v_1v_2, e_2 = v_2v_3, e_3 = v_3v_1\}$.

   • Each vertex in $G$ has degree 2, so $G$ is Eulerian.

2. Line graph $L(G)$:

   • The vertices of $L(G)$ correspond to edges of $G$: $\{e_1, e_2, e_3\}$.

   • Two vertices in $L(G)$ are adjacent if their corresponding edges in $G$ share a common vertex.

   • In this case, $L(G)$ is also a triangle because every pair of edges in $G$ shares a vertex.

   • $L(G)$ is Eulerian because all vertices in $L(G)$ have degree 2.

---

Conclusion

Both $G$ and $L(G)$ are Eulerian, satisfying the problem's condition.

---

Would you like more detailed diagrams or steps? Here are some related questions for exploration:

1. What is the definition of a line graph and how is it constructed for general graphs?
2. Can we construct a non-Eulerian $G$ such that $L(G)$ is Eulerian?
3. How does the degree of a vertex in $G$ relate to the degree of its corresponding vertices in $L(G)$?
4. What is the smallest graph where $L(G)$ is not connected?
5. Are there any practical applications of Eulerian graphs in real-world problems?

Tip: To confirm if $L(G)$ is Eulerian, always check the degree of its vertices, as it directly depends on the edge-sharing properties of $G$.