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 @d(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|= =3. The conditional edge-fault tolerant hamiltonian connectivity HC"e^3(G) 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 K"n to denote the complete graph with n vertices. In this paper, we show that HC"e^3(K"n)=2n-10 for n@__ __{4,5,8,10}, HC"e^3(K"4)=0, HC"e^3(K"5)=2, HC"e^3(K"8)=5, and HC"e^3(K"1"0)=9.