On n-Hamiltonian line graphs

With each nonempty graph G one can associate a graph L(G), called the line graph of G, with the property that there exists a one-to-one correspondence between E(G) and V(L(G)) such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. For integers 171 > 2, the FntR iterated line graph L”(G) of G is defined to be L(P-l(G)). A graph G of order p > 3 is n-Hamiltonian, 0 < N < p 3, if the removal of any k vertices, 0 < k < n, resu!ts in a Hamiltonian graph. It is shown that if G is a connected graph wirb 6(G) >, 3, where S(G) denotes the minimum degree of 6, then L”(G) is (6(G) 3)-Mamiltonian. Furthermore, if G is 2-connected and 6(G) > 4, then P(G) is (26(G) 4)-Hamiltonian. For a connected graph G which is neither a path, a cycle, nor the graph K&3) and for any positive integer n, the existence of an integer k such that LI”(G) is n-Hamiltonian for every wz > k is exhibited. Then, for the special case PI = 1, bounds on (and, in some cases, the exact vatue of) the smallest such integer k are determined for various classes of graphs.