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Definition:

Let $G$ be a graph, the line graph of $G$ denoted of $L(G)$ is defined as follows:

1-The vertices of $L(G)$ are the edges of $G$

2-Two vertices of $L(G)$ are adjacent iff their corresponding edges in $G$ are incident.

Question?

-It is well known that if $G$ Hamiltonian or Eulerian then $L(G)$ is also Hamiltonian (see wikipedia).

-My question is the inverse is also correct in certain cases/ under certain conditions?

(for example, if $L(G)$ is Hamiltonian and if there is a certain condition on $G$ "which I do not know yet" will this proves that $G$ is also Hamiltonian?).

Any idea will be useful!

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