Non-traceability of large connected claw-free graphs

Abstract

Let G be a connected claw-free graph on n vertices. Let ζ3(G) be the minimum degree sum among triples of independent vertices in G. It is proved that if ζ3(G) ≥ n - 3 then G is traceable or else G is one of graphs Gn each of which comprises three disjoint nontrivial complete graphs joined together by three additional edges which induce a triangle K3. Moreover, it is shown that for any integer k ≥ 4 there exists a positive integer V(k) such that if ζ3(G) ≥ n - k, n > V((k) and G is non-traceable, then G is a factor of a graph Gn. 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 ζ3(G) ≥ $\frac{5}{6}|V| - 3$. This contrasts sharply with known results on NP-completeness among dense graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 7586, 1998