Clique covering and degree conditions for hamiltonicity in claw-free graphs Original Research Article

By using the closure concept introduced by the last author, we prove that for any sufficiently large nonhamiltonian claw-free graph G satisfying a degree condition of type σk(G)>n+k2−4k+7 (where k is a constant), the closure of G can be covered by at most k−1 cliques. Using structural properties of the closure concept, we show a method for characterizing all such nonhamiltonian exceptional graphs with limited clique covering number. The method is explicitly carried out for k⩽6 and illustrated by proving that every 2-connected claw-free graph G of order n⩾77 with δ(G)⩾14 and σ6(G)>n+19 is either hamiltonian or belongs to a family of easily described exceptions.