An s -Hamiltonian Line Graph Problem

For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In [J. Graph Theory, 11 (1987), 399-407], Broersma and Veldman proposed an open problem: For a given positive integer k, determine the value s for which the statement Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, that if the statement holds when s = 2k. In this paper, we proved that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.