Graphs in this question are finite, simple and undirected. A function $\psi\colon V(G)\to V(H)$ is a locally bijective homomorphism from $G$ to $H$ if for every vertex $v$ of $G$, the restiction of $\psi$ to $N_G(v)$ is a bijection from $N_G(v)$ onto $N_H(\psi(v))$ (observe that such a function $\psi$ is always a homomorphism from $G$ to $H$). An alternate defintion is given below.
See [1] for a suvrey on locally constrained graph homomorphisms.
If $G$ admits a locally bijective homomorphism to $H$, then clearly the line graph of $G$ admits a locally bijective homomorphism to the line graph of $H$. Is the converse true?
If the line graph of $G$ admits a locally bijective homomorphism to the line graph of $H$, can we say that $G$ admits a locally bijective homomorphism to $H$?
Definitions
A homomorphism from a graph $G$ to a graph $H$ is a fuction $\psi\colon V(G)\to V(H)$ such that $\psi(u)\psi(v)\in E(H)$ whenever $uv\in E(G)$.
A function $\psi\colon V(G)\to V(H)$ is a locally bijective homomorphism from $G$ to $H$ if for every vertex $v$ of $G$, the restiction of $\psi$ to $N_G(v)$ is a bijection from $N_G(v)$ onto $N_H(\psi(v))$ (such a function $\psi$ is always a homomorphism from $G$ to $H$).
In other words, a homomorphism $\psi$ from $G$ to $H$ is locally bijective if if (i) every vertex $v$ of $G$ mapped by $\psi$ to a vertex $w$ of $H$ is said to be a copy of $w$ in $G$, and (ii) for each vertex $w$ of $H$ and each neighbour $x$ of $w$ in $H$, each copy of $w$ in $G$ has exactly one copy of $x$ in $G$ as its neighbour.
In the figure above, vertices in $H$ are labelled distinct and are drawn by distinct shapes, whereas each copy of $w$ in $G$ has the same label and shape as $w$.
Reference
[1] Fiala, Jiří; Kratochvíl, Jan, Locally constrained graph homomorphisms – structure, complexity, and applications, Comput. Sci. Rev. 2, No. 2, 97-111 (2008). ZBL1302.05122.
