If $G = (V,E)$ is a finite, connected, simple, undirected graph, is there a [Hamiltonian path](https://en.wikipedia.org/wiki/Hamiltonian_path) in the [line graph](https://en.wikipedia.org/wiki/Line_graph) $L(G)$ of $G$?



Hamiltonian path in the line graph of a connected graph
No. Consider the graph $G = (V, E)$ with
$$V = \{0,1,2,3,4,5,6\}\quad \text{and} \quad E = \{\{0,1\}, \{1, 2\}, \{0,3\}, \{3, 4\}, \{0,5\}, \{5, 6\}\}.$$
Note that its line-graph has three vertices of degree 1. Therefore $L(G)$ has no Hamiltonian path.


