# Locally bijective homomorphism between line graphs

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.