Conditional fault hamiltonian connectivity of the complete graph

A path in G is a hamiltonian path if it contains all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path between any two distinct vertices of G. The degree of a vertex u in G is the number of vertices of G adjacent to u. We denote by δ(G) the minimum degree of vertices of G. A graph G is conditional k edge-fault tolerant hamiltonian connected if G−F is hamiltonian connected for every F⊂E(G) with |F|⩽k and δ(G−F)⩾3. The conditional edge-fault tolerant hamiltonian connectivity is defined as the maximum integer k such that G is k edge-fault tolerant conditional hamiltonian connected if G is hamiltonian connected and is undefined otherwise. Let n⩾4. We use Kn to denote the complete graph with n vertices. In this paper, we show that for n∉{4,5,8,10}, , , , and .