A graph is k-triangular if each edge is in at least k triangles. Triangular is a synonym for 1-triangular. It is shown that the line graph of a triangular graph of order at least 4 is panconnected if and only if it is 3-connected. Furthermore, the line graph of a k-triangular graph is k-hamiltonian if and only if it is (k + 2)-connected (k ≥ 1). These results generalize work of Clark and Wormald and of Lesniak-Foster. Related results are due to Oberly and Sumner and to Kanetkar and Rao.
Broersma, H. J., & Veldman, H. J. (1987). 3-Connected line graphs of triangular graphs are panconnected and 1-hamiltonian. Journal of graph theory, 11(3), 399-407. https://doi.org/10.1002/jgt.3190110314