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.
-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!