Non-traceability of large connected claw-free graphs

Let G be a connected claw-free graph on n vertices. Let sigma(3)(G) be the minimum degree sum among triples of independent vertices in G. It is proved that if sigma(3)(G) greater than or equal to n-3 then G is traceable or else G is one of graphs G, each of which comprises three disjoint nontrivial complete graphs joined together by three additional edges which induce a triangle K-3 Moreover, it is shown that for any integer k greater than or equal to 4 there exists a positive integer nu(k) such that if sigma(3)(G) greater than or equal to n - k, n > nu(k) and G is non-traceable, then G is a factor of a graph G,. Consequently, the problem HAMILTONIAN PATH restricted to claw-free graphs G (V, E) (which is known to be NP-complete) has linear time complexity O(\E\) provided that sigma(3)(G) greater than or equal to 5/6\V\ - 3. This contrasts sharply with known results on NP-completeness among dense graphs. (C) 1998 John Wiley & Sons,Inc.